4 Concepts10 Practice Qs30 Mock Qs⚡ Desmos Speed Hacks
Key Concepts & Worked Archetypes
Concept 1
Concept 1: Slope-Intercept Form & Interpretation
A linear equation in two variables represents a straight-line relationship where the rate of change is constant.
Standard slope-intercept form is given by $y = mx + b$
The slope $m$ represents the constant rate of change: $m = \frac{y_2 - y_1}{x_2 - x_1}$
The y-intercept $b$ represents the initial value when $x = 0$
📘 Traditional Algebraic Method
Identify two coordinate points $(x_1, y_1)$ and $(x_2, y_2)$ from the given text or table.
⚡ SAT Speed Trick & Desmos Hack
Type the two points directly into Desmos as a table or use regression $y_1 \sim m x_1 + b$ to instantly extract slope and intercept.
💡 Worked SAT Archetype Example
Problem: A linear function $f$ models the value of a machine, in dollars, $x$ years after purchase. If $f(2) = 18000$ and $f(5) = 12000$, what is the value of the machine 7 years after purchase?
📘 Step-by-Step Textbook Solution:
Step 1
Set up the points as coordinates: $(2, 18000)$ and $(5, 12000)$
Simplify the slope calculation: $m = \frac{-6000}{3} = -2000$
Step 4
Write the equation: $f(x) = -2000x + b$
Step 5
Solve for $b$ using $(2, 18000)$: $18000 = -2000(2) + b$
Step 6
Determine $b$: $b = 22000$
Step 7
Evaluate for $x = 7$: $f(7) = -2000(7) + 22000$
Step 8
Final calculation and answer: $f(7) = 8000$
⚡ Speed / Desmos Tactic:
Step 1
Open Desmos and create a table with columns $x_1$ and $y_1$, entering $(2, 18000)$ and $(5, 12000)$
Step 2
Type the regression command: $y_1 \sim mx_1 + b$
Step 3
Read values $m = -2000$ and $b = 22000$, then evaluate $-2000(7) + 22000$ to get 8000
Concept 2
Concept 2: Standard Form & Intercepts
Standard form provides a symmetric way to express linear relations and quickly find axis intersections.
Standard form is expressed as $Ax + By = C$, where $A$, $B$, and $C$ are integers.
The x-intercept occurs where $y = 0$, calculated as $x = \frac{C}{A}$.
The y-intercept occurs where $x = 0$, calculated as $y = \frac{C}{B}$.
📘 Traditional Algebraic Method
Substitute $x = 0$ to find the y-intercept and $y = 0$ to find the x-intercept algebraically.
⚡ SAT Speed Trick & Desmos Hack
Type the equation directly into Desmos and click the highlighted x-intercept and y-intercept icons on the coordinate axes.
💡 Worked SAT Archetype Example
Problem: What is the x-intercept of the graph of $3x - 4y = 24$ in the xy-plane?
📘 Step-by-Step Textbook Solution:
Step 1
Set $y = 0$ to find the x-intercept.
Step 2
Substitute into the equation: $3x - 4(0) = 24$
Step 3
Simplify the expression: $3x = 24$
Step 4
Solve for $x$: $x = 8$
⚡ Speed / Desmos Tactic:
Step 1
Input $3x - 4y = 24$ into the Desmos graphing window.
Step 2
Click on the point where the line crosses the x-axis.
Step 3
Read coordinate $(8, 0)$ instantly.
Concept 3
Concept 3: Word Problems & Contextual Parameters
Contextual linear equations translate real-world scenarios into mathematical models with fixed costs and variable rates.
The coefficient of the variable represents the unit rate or cost per item.
The constant term represents the fixed fee, starting value, or base price.
Ensure units match across all terms in the equation before solving.
📘 Traditional Algebraic Method
Define variables for unknown quantities, construct an equation matching the word description, and solve step-by-step.
⚡ SAT Speed Trick & Desmos Hack
Use Desmos sliders or plug options from multiple-choice questions directly into the defined expression.
💡 Worked SAT Archetype Example
Problem: A phone company charges a monthly base fee of $15 plus $0.05 per text message sent. If a customer's bill is $28.50, how many text messages were sent?
📘 Step-by-Step Textbook Solution:
Step 1
Define variable $t$ as the number of text messages sent.
Step 2
Set up the equation: $0.05t + 15 = 28.50$
Step 3
Subtract 15 from both sides: $0.05t = 13.50$
Step 4
Divide by 0.05: $t = 270$
⚡ Speed / Desmos Tactic:
Step 1
Type $y = 0.05x + 15$ into Desmos.
Step 2
Type $y = 28.50$ as a second equation.
Step 3
Click the intersection point to read $x = 270$.
Concept 4
Concept 4: Parallel and Perpendicular Lines
Geometric relationships between linear equations are determined exclusively by comparing their slopes.
Vertical lines have undefined slope ($x = c$), while horizontal lines have zero slope ($y = c$).
📘 Traditional Algebraic Method
Extract slope $m$ from the given equation, apply the slope rule for parallel or perpendicular lines, and use point-slope form.
⚡ SAT Speed Trick & Desmos Hack
Use Desmos to visually verify perpendicularity or parallelism by graphing the reference line and testing answer choices.
💡 Worked SAT Archetype Example
Problem: Which of the following equations represents a line that is perpendicular to the graph of $2x + 3y = 6$?
📘 Step-by-Step Textbook Solution:
Step 1
Rewrite $2x + 3y = 6$ in slope-intercept form.
Step 2
Isolate $y$: $3y = -2x + 6$
Step 3
Simplify: $y = -\frac{2}{3}x + 2$
Step 4
Identify slope $m = -\frac{2}{3}$.
Step 5
Find the negative reciprocal: $m_{\perp} = \frac{3}{2}$.
⚡ Speed / Desmos Tactic:
Step 1
Graph $2x + 3y = 6$ in Desmos.
Step 2
Test answer choices looking for visually 90-degree intersecting lines, ensuring slopes match $\frac{3}{2}$.
Practice Questions (10)
Question 1Archetype
Medium
Hint: The slope is the coefficient of the independent variable $n$.
📘 Step-by-Step Algebraic Solution
Step 1
Locate the slope in the equation $P = 6.50n - 120$, which is $6.50$.
Step 2
Note that $n$ represents the number of candles sold.
Step 3
Determine that for each increase of $1$ in $n$, the profit $P$ increases by $6.50$.
Step 4
Conclude that $6.50$ is the profit earned per candle sold.
⚡ Desmos Shortcut / Speed Hack
Step 1
Look at the number multiplying the variable $n$.
Step 2
Multipliers of independent variables represent unit rates.
Step 3
Select option A.
Question 9Interpreting Slope and Intercepts in Word Problems
Medium
A pharmaceutical representative's monthly salary $S$, in dollars, is determined by a base salary plus a commission for every prescription software package sold, $p$. The equation is $S = 2500 + 150p$. If the representative's salary increases by $\$450$, how many more software packages were sold?
Hint: Divide the increase in salary by the commission rate per package sold.
📘 Step-by-Step Algebraic Solution
Step 1
Identify the commission rate per package sold as the slope, $150$ dollars per package.
Step 2
Set up the equation relating salary increase to packages sold: $\Delta S = 150 \cdot \Delta p$.
Step 3
Substitute the given salary increase: $450 = 150 \cdot \Delta p$.
Step 4
Solve for $\Delta p$: $\Delta p = \frac{450}{150} = 3$ packages.
⚡ Desmos Shortcut / Speed Hack
Step 1
Compute $\frac{\text{Total Increase}}{\text{Rate per unit}}$.
Step 2
$\frac{450}{150} = 3$.
Step 3
Select option B.
Question 10Interpreting Slope and Intercepts in Word Problems
Hard
A runner's distance $d$, in miles, from a hydration checkpoint after $t$ minutes is modeled by $d = 5.2 - 0.15t$. Which of the following statements is the best interpretation of both the slope and the y-intercept in this context?
Hint: A positive intercept means starting distance, and a negative slope means distance is decreasing over time.
📘 Step-by-Step Algebraic Solution
Step 1
Analyze the y-intercept $5.2$: at $t = 0$, $d = 5.2$, meaning the initial distance from the checkpoint is $5.2$ miles.
Step 2
Analyze the slope $-0.15$: distance decreases by $0.15$ miles per minute as time increases.
Step 3
Because distance from the checkpoint is decreasing over time, the runner is moving closer to the checkpoint.
Step 4
Combine these findings to confirm option A is correct.
⚡ Desmos Shortcut / Speed Hack
Step 1
Negative slope implies distance is shrinking, meaning moving *toward* the target.
Step 2
Intercept $5.2$ is the starting distance.
Step 3
Instantly select option A.
Question 11Writing Linear Equations from Two Points or Data
Easy
A linear function passes through the points $(0, 3)$ and $(2, 7)$ in the $xy$-plane. Which of the following is the equation of this line?
Hint: The point $(0, 3)$ directly gives you the y-intercept $b$. Find the slope $m$ using the two points formula.
📘 Step-by-Step Algebraic Solution
Step 1
Identify the y-intercept $b = 3$ from the point $(0, 3)$.
Step 2
Calculate the slope $m$ using points $(0, 3)$ and $(2, 7)$: $m = \frac{7 - 3}{2 - 0} = \frac{4}{2} = 2$.
Step 3
Substitute $m = 2$ and $b = 3$ into slope-intercept form $y = mx + b$.
Step 4
Obtain the equation $y = 2x + 3$.
⚡ Desmos Shortcut / Speed Hack
Step 1
Test point $(0, 3)$ in the options: $3 = 2(0) + 3$ (True for A, C).
Step 2
Test point $(2, 7)$ in option A: $7 = 2(2) + 3 = 7$ (True).
Step 3
Select option A.
Question 12Writing Linear Equations from Two Points or Data
Easy
A line in the $xy$-plane has a slope of $-3$ and passes through the point $(1, 4)$. Which of the following is the equation of the line?
Hint: Use the point-slope form $y - y_1 = m(x - x_1)$ or substitute the point into $y = -3x + b$ to find $b$.
📘 Step-by-Step Algebraic Solution
Step 1
Start with the slope-intercept form $y = -3x + b$.
Step 2
Substitute the coordinates of the given point $(1, 4)$ for $x$ and $y$: $4 = -3(1) + b$.
Question 15Writing Linear Equations from Two Points or Data
Hard
Line $k$ is perpendicular to the line $2x + 5y = 10$ and passes through the point $(-4, 1)$. What is the y-intercept of line $k$?
Hint: Perpendicular lines have negative reciprocal slopes. Find the slope of the given line first.
📘 Step-by-Step Algebraic Solution
Step 1
Find the slope of the given line $2x + 5y = 10$ by converting to slope-intercept form: $5y = -2x + 10 \implies y = -\frac{2}{5}x + 2$.
Step 2
Identify the slope as $-\frac{2}{5}$.
Step 3
Determine the slope of line $k$ as the negative reciprocal: $m_k = \frac{5}{2}$.
Step 4
Use point-slope form with point $(-4, 1)$: $y - 1 = \frac{5}{2}(x - (-4))$.
Step 5
Simplify: $y - 1 = \frac{5}{2}x + 10 \implies y = \frac{5}{2}x + 11$, wait let's recompute: $5/2 \times 4 = 10$, so $y - 1 = 5/2 x + 10 \implies y = 5/2 x + 11$. Wait, let's check $(-4, 1)$: $\frac{5}{2}(-4) + 11 = -10 + 11 = 1$. Correct y-intercept is $(0, 11)$? Let's check options: Option D is $(0, 11)$. Let's re-verify: $y = \frac{5}{2}x + 11$ gives y-intercept $11$. Let's change correct index to D.
⚡ Desmos Shortcut / Speed Hack
Step 1
Slope of given line is $-2/5$, so perpendicular slope is $5/2$.
Step 2
Write equation: $y = \frac{5}{2}x + b$, substitute $(-4, 1)$: $1 = \frac{5}{2}(-4) + b \implies 1 = -10 + b \implies b = 11$.
Step 3
Select option D.
Question 16Parallel and Perpendicular Lines in Linear Equations
Easy
Which of the following lines is parallel to the graph of $y = 4x - 7$?
Hint: Parallel lines share the exact same slope.
📘 Step-by-Step Algebraic Solution
Step 1
Identify the slope of the given line $y = 4x - 7$, which is $m = 4$.
Step 2
Recall that parallel lines must have identical slopes.
Step 3
Inspect the options for a line with a slope of $4$.
Step 4
Option B has the equation $y = 4x + 5$, which also has a slope of $4$.
⚡ Desmos Shortcut / Speed Hack
Step 1
Look for the coefficient of $x$ to be $4$.
Step 2
Only option B has a slope of $4$.
Step 3
Select option B.
Question 17Parallel and Perpendicular Lines in Linear Equations
Easy
Which of the following equations represents a line that is perpendicular to $y = -\frac{2}{3}x + 5$?
Hint: Perpendicular lines have slopes that are negative reciprocals of each other.
📘 Step-by-Step Algebraic Solution
⚡ Desmos Shortcut / Speed Hack
Official SAT Exam Blueprint & Weightage
Metric
Exam Pattern & Weightage
Question Frequency
1-3 questions per test module
Module 1 Appearance
Medium Frequency (Foundational tests)
Module 2 Appearance
High Frequency (Hard module score differentiator)
Pattern Analysis
Analysis of College Board question patterns and recent exam distributions.
🏛️
Official SAT PYQ Drill Bank (2023–2026)
We are compiling real verified College Board exam patterns with step-by-step Desmos shortcuts for Linear Equations Two Variables.
Updated for 2026 Testing Season
Revision, Traps & Desmos Syntax
Slope Formula
$m = \frac{y_2 - y_1}{x_2 - x_1}$
Used to find the rate of change between two coordinate points.
Slope-Intercept Form
$y = mx + b$
Core equation format where $m$ is slope and $b$ is the y-intercept.
Point-Slope Form
$y - y_1 = m(x - x_1)$
Fastest formula when given a point and a slope.
🚨 Top SAT Traps & Misconceptions
⚠️ SAT Trap: Confusing X and Y Intercepts
College Board frequently asks for the x-intercept, but students automatically solve for the y-intercept (setting $x=0$ instead of $y=0$).
⚠️ SAT Trap: Unit Mismatch in Word Problems
Rates given in cents while answers require dollars (or minutes vs. hours). Always verify unit consistency.
⚡ Essential Desmos Cheatsheet
🎯 Linear Regression Table
y_1 ~ m x_1 + b
Instantly find linear equations from any two data points or coordinate pairs.
🎯 Intersection Solver
Type both equations as separate lines
Click the intersection point to get exact coordinate values without simultaneous algebraic solving.
3-Level Mock Test (30 Questions)
🟢 Level 1: Foundation
10 Qs · Sub-600 Score
🟡 Level 2: Target 700+
10 Qs · 600–740 Score
🔴 Level 3: 800-Mastery
10 Qs · 750–800 Score
Question 1Level 1: Foundation
What is the slope of the line represented by the equation $y = 3x - 5$?
Explanation:
Step 1
Identify the slope-intercept form of a linear equation, which is $y = mx + b$, where $m$ is the slope and $b$ is the $y$-intercept.
Step 2
Compare the given equation $y = 3x - 5$ with the slope-intercept form to find that the coefficient of $x$ is $3$.
Step 3
Conclude that the slope $m$ is $3$, which matches option A.
Question 2Level 1: Foundation
What is the $y$-intercept of the line given by the equation $2x + 4y = 12$?
Explanation:
Step 1
Find the $y$-intercept by setting $x = 0$ in the given equation $2x + 4y = 12$.
Step 2
Substitute $x = 0$ to get $2(0) + 4y = 12$, which simplifies to $4y = 12$.
Step 3
Solve for $y$ by dividing both sides by $4$, yielding $y = 3$, so the $y$-intercept is $(0, 3)$.
Question 3Level 1: Foundation
Which of the following points lies on the graph of the line $y = -2x + 7$?
Explanation:
Step 1
Test each given point by substituting its $x$-coordinate into the equation $y = -2x + 7$ to see if the resulting $y$-coordinate matches.
Step 2
Test point $(2, 3)$: $y = -2(2) + 7 = -4 + 7 = 3$, which matches the given $y$-coordinate.
Step 3
Conclude that the point $(2, 3)$ lies on the line, corresponding to option B.
Question 4Level 1: Foundation
A line passes through the points $(0, 1)$ and $(2, 5)$. What is the slope of this line?
Explanation:
Step 1
Use the slope formula $m = \frac{y_2 - y_1}{x_2 - x_1}$ for two points $(x_1, y_1)$ and $(x_2, y_2)$.
Step 2
Substitute the given points $(0, 1)$ and $(2, 5)$ into the formula: $m = \frac{5 - 1}{2 - 0}$.
Step 3
Simplify the fraction: $m = \frac{4}{2} = 2$, which corresponds to option B.
Question 5Level 1: Foundation
Which equation represents a line that is parallel to the graph of $y = 5x - 2$?
Explanation:
Step 1
Recall that parallel lines share the exact same slope.
Step 2
Identify the slope of the given line $y = 5x - 2$, which is $5$.
Step 3
Look for an equation among the options with a slope of $5$, which is $y = 5x + 8$ (option B).
Question 6Level 1: Foundation
What is the $x$-intercept of the linear equation $3x - 2y = 12$?
Explanation:
Step 1
To find the $x$-intercept, set $y = 0$ in the equation $3x - 2y = 12$.
Step 2
Substitute $y = 0$ to get $3x - 2(0) = 12$, which simplifies to $3x = 12$.
Step 3
Solve for $x$ by dividing by $3$, yielding $x = 4$, so the $x$-intercept is $(4, 0)$.
Question 7Level 1: Foundation
If a line has a slope of $-3$ and passes through the point $(0, 4)$, what is its equation in slope-intercept form?
Explanation:
Step 1
Use the slope-intercept form $y = mx + b$.
Step 2
Substitute the given slope $m = -3$ and the $y$-intercept value $b = 4$ (from point $(0, 4)$).
Step 3
Write the final equation: $y = -3x + 4$, matching option B.
Question 8Level 1: Foundation
Which of the following ordered pairs is a solution to the system of equations $\begin{cases} x + y = 5 \\ x - y = 1 \end{cases}$?
Explanation:
Step 1
Add the two equations together using elimination: $(x + y) + (x - y) = 5 + 1$.
Step 2
Simplify to get $2x = 6$, which yields $x = 3$.
Step 3
Substitute $x = 3$ back into $x + y = 5$ to find $y = 2$, giving the solution $(3, 2)$, which is option C.
Question 9Level 1: Foundation
What is the slope of any vertical line in the coordinate plane?
Explanation:
Step 1
Recall that a vertical line has an equation of the form $x = c$, where $c$ is a constant.
Step 2
For any two points on a vertical line, the $x$-coordinates are identical, meaning the change in $x$ ($\Delta x$) is $0$.
Step 3
Since the slope formula involves division by $\Delta x$, dividing by zero makes the slope undefined.
Question 10Level 1: Foundation
Convert the equation $2x - y = 4$ into slope-intercept form.
Explanation:
Step 1
Start with the given equation $2x - y = 4$.
Step 2
Subtract $2x$ from both sides to isolate the $y$-term: $-y = -2x + 4$.
Step 3
Multiply both sides by $-1$ to solve for $y$: $y = 2x - 4$, which corresponds to option A.
Question 1Level 2: Target 700+
Line $l$ is represented by the equation $3x + 4y = 12$. Line $k$ is perpendicular to line $l$ and passes through the origin. What is the equation of line $k$?
Explanation:
Step 1
Rewrite $3x + 4y = 12$ in slope-intercept form: $4y = -3x + 12$, so $y = -\frac{3}{4}x + 3$.
Step 2
Identify the slope of line $l$ as $-\frac{3}{4}$. The slope of a perpendicular line is the negative reciprocal, which is $\frac{4}{3}$.
Step 3
Since line $k$ passes through the origin $(0, 0)$, its $y$-intercept is $0$, yielding the equation $y = \frac{4}{3}x$ (option B).
Question 2Level 2: Target 700+
In the $xy$-plane, a line passes through the points $(-2, 5)$ and $(4, k)$. If the slope of the line is $-\frac{1}{2}$, what is the value of $k$?
Explanation:
Step 1
Apply the slope formula with the given points $(-2, 5)$ and $(4, k)$, and set it equal to $-\frac{1}{2}$: $\frac{k - 5}{4 - (-2)} = -\frac{1}{2}$.
Step 2
Simplify the denominator: $\frac{k - 5}{6} = -\frac{1}{2}$.
Step 3
Multiply both sides by $6$ to get $k - 5 = -3$, and solve for $k$ to find $k = 2$ (option A).
Question 3Level 2: Target 700+
Line $p$ has a slope of $3$ and passes through the point $(2, 7)$. Which of the following points also lies on line $p$?
Explanation:
Step 1
Write the equation of line $p$ in point-slope form using point $(2, 7)$ and slope $3$: $y - 7 = 3(x - 2)$.
Step 2
Convert to slope-intercept form: $y - 7 = 3x - 6 \implies y = 3x + 1$.
Step 3
Test the options. For $(1, 4)$: $y = 3(1) + 1 = 4$, which is correct, so option D is the right answer.
Question 4Level 2: Target 700+
For what value of $k$ will the system of linear equations $\begin{cases} 2x + 3y = 7 \\ 4x + ky = 14 \end{cases}$ have infinitely many solutions?
Explanation:
Step 1
Recall that a linear system has infinitely many solutions when the two equations are identical multiples of one another.
Step 2
Multiply the first equation $2x + 3y = 7$ by $2$ to match the $x$-coefficient of the second equation: $4x + 6y = 14$.
Step 3
Compare this to $4x + ky = 14$ to conclude that $k = 6$, corresponding to option B.
Question 5Level 2: Target 700+
What is the area of the triangle formed by the coordinate axes and the line $3x + 4y = 12$?
Explanation:
Step 1
Find the $x$-intercept by setting $y = 0$, giving $3x = 12 \implies x = 4$, so the base is $4$.
Step 2
Find the $y$-intercept by setting $x = 0$, giving $4y = 12 \implies y = 3$, so the height is $3$.
Step 3
Calculate the area of the right triangle: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 4 \times 3 = 6$ (option A).
Question 6Level 2: Target 700+
A system of two linear equations has no solution. If the first equation is $5x - 2y = 8$, which of the following could be the second equation?
Explanation:
Step 1
Understand that a system with no solution represents parallel lines that have identical slopes but different $y$-intercepts.
Step 2
Rewrite $5x - 2y = 8$ in slope-intercept form: $y = \frac{5}{2}x - 4$, which has a slope of $\frac{5}{2}$ and a $y$-intercept of $-4$.
Step 3
Examine option C: $-10x + 4y = 12 \implies 4y = 10x + 12 \implies y = \frac{5}{2}x + 3$. This has the same slope $\frac{5}{2}$ but a different $y$-intercept ($3$ vs $-4$), making the lines parallel with no solution.
Question 7Level 2: Target 700+
The line $y = mx + b$ passes through $(-1, 2)$ and is parallel to the line $4x - 2y = 6$. What is the value of $b$?
Explanation:
Step 1
Find the slope of the given line $4x - 2y = 6$ by rewriting it in slope-intercept form: $-2y = -4x + 6 \implies y = 2x - 3$.
Step 2
The parallel line shares the same slope $m = 2$, so the equation is $y = 2x + b$.
Step 3
Substitute the point $(-1, 2)$ into $y = 2x + b$: $2 = 2(-1) + b \implies 2 = -2 + b \implies b = 4$ (option B).
Question 8Level 2: Target 700+
If $3x - 5y = 15$, what is the value of $\frac{y}{x}$ when $x \neq 0$ and $y \neq 0$?
Explanation:
Step 1
Rearrange the equation $3x - 5y = 15$ to solve for $5y$: $5y = 3x - 15$.
Step 2
Divide both sides by $5x$ to find $\frac{y}{x}$: $\frac{y}{x} = \frac{3x - 15}{5x}$.
Step 3
Simplify the expression: $\frac{3x}{5x} - \frac{15}{5x} = \frac{3}{5} - \frac{3}{x}$, which matches option A.
Question 9Level 2: Target 700+
Line $m$ passes through $(2, 3)$ and $(6, 11)$. Line $n$ is perpendicular to line $m$. What is the slope of line $n$?
Explanation:
Step 1
Calculate the slope of line $m$ using the points $(2, 3)$ and $(6, 11)$: $m = \frac{11 - 3}{6 - 2} = \frac{8}{4} = 2$.
Step 2
Recall that perpendicular lines have slopes that are negative reciprocals of each other.
Step 3
Take the negative reciprocal of $2$ to find the slope of line $n$: $-\frac{1}{2}$ (option C).
Question 10Level 2: Target 700+
A linear function $f$ satisfies $f(2) = 5$ and $f(5) = 14$. What is the value of $f(10)$?
Explanation:
Step 1
Find the slope of the linear function using points $(2, 5)$ and $(5, 14)$: $m = \frac{14 - 5}{5 - 2} = \frac{9}{3} = 3$.
Step 2
Write the equation for the function: $f(x) = 3(x - 2) + 5 \implies f(x) = 3x - 6 + 5 \implies f(x) = 3x - 1$.
Step 3
Evaluate the function at $x = 10$: $f(10) = 3(10) - 1 = 30 - 1 = 29$, corresponding to option B.
Question 1Level 3: 800 Mastery
In the $xy$-plane, line $l$ has the equation $ax + by = c$, where $a$, $b$, and $c$ are positive constants. Which of the following conditions ensures that line $l$ has a positive slope and a negative $y$-intercept?
Explanation:
Step 1
Convert $ax + by = c$ to slope-intercept form: $by = -ax + c \implies y = -\frac{a}{b}x + \frac{c}{b}$.
Step 2
The slope is $-\frac{a}{b}$. For the slope to be positive, we must have $-\frac{a}{b} > 0$, meaning $a$ and $b$ must have opposite signs.
Step 3
The $y$-intercept is $\frac{c}{b}$. For the $y$-intercept to be negative when $b$ is negative, $c$ must be positive (since a positive divided by a negative is negative). Thus, $a > 0, b < 0, c > 0$ (option C).
Question 2Level 3: 800 Mastery
Line $j$ passes through the points $(2, 5)$ and $(6, 13)$. Line $k$ is perpendicular to line $j$ and intersects it at its $y$-intercept. What is the $x$-intercept of line $k$?
Explanation:
Step 1
Find the slope of line $j$: $m_j = \frac{13 - 5}{6 - 2} = \frac{8}{4} = 2$.
Step 2
Find the equation of line $j$ using point $(2, 5)$: $y - 5 = 2(x - 2) \implies y = 2x + 1$. The $y$-intercept is $(0, 1)$.
Step 3
Line $k$ is perpendicular to line $j$, so its slope is $m_k = -\frac{1}{2}$. Since it passes through $(0, 1)$, its equation is $y = -\frac{1}{2}x + 1$. Set $y = 0$ to find the $x$-intercept: $0 = -\frac{1}{2}x + 1 \implies x = 2$, wait, let's recalculate: $0 = -\frac{1}{2}x + 1 \implies \frac{1}{2}x = 1 \implies x = 2$. Let's re-verify options. Wait, if $y = -\frac{1}{2}x + 1$, then $x$-intercept is $(2, 0)$. Let's check option C: $2.5$. Wait, let's re-read: slope $m_j = 2$. $y$-intercept is $(0, 1)$. Let's re-check $y = 2(0) + 1 = 1$. Perpendicular slope is $-\frac{1}{2}$. Equation of $k$: $y - 1 = -\frac{1}{2}(x - 0) \implies y = -\frac{1}{2}x + 1$. Set $y = 0$: $0 = -\frac{1}{2}x + 1 \implies x = 2$. Wait, option B is $(2, 0)$. Let's check if the correct index is B.
Question 3Level 3: 800 Mastery
If the system of linear equations $\begin{cases} kx + 3y = 4 \\ 4x + 6y = 8 \end{cases}$ has infinitely many solutions, what is the value of $k$?
Explanation:
Step 1
For a system of linear equations to have infinitely many solutions, the two equations must be proportional.
Step 2
Compare the second equation $4x + 6y = 8$ with the first equation $kx + 3y = 4$ by observing that dividing the second equation by $2$ gives $2x + 3y = 4$.
Step 3
Matching coefficients of $x$ gives $k = 2$, corresponding to option A.
Question 4Level 3: 800 Mastery
A line in the $xy$-plane passes through the origin and has a positive slope $m$. If the line passes through the point $(a, b)$ where $a > 0$ and $b > 0$, which of the following must be equal to $m$?
Explanation:
Step 1
Use the slope formula for a line passing through the origin $(0, 0)$ and a point $(a, b)$.
Step 2
Substitute the coordinates into the formula: $m = \frac{b - 0}{a - 0} = \frac{b}{a}$.
Step 3
Conclude that the slope $m$ is equal to $\frac{b}{a}$, which matches option B.
Question 5Level 3: 800 Mastery
In the $xy$-plane, line $1$ has equation $y = 3x - 2$. Line $2$ is the reflection of line $1$ across the line $y = x$. What is the equation of line $2$?
Explanation:
Step 1
Recall that reflecting a graph across the line $y = x$ is equivalent to finding the inverse function by swapping $x$ and $y$.
Step 2
Swap $x$ and $y$ in the equation $y = 3x - 2$ to get $x = 3y - 2$.
Step 3
Solve for $y$: $3y = x + 2 \implies y = \frac{1}{3}x + \frac{2}{3}$, which corresponds to option B.
Question 6Level 3: 800 Mastery
If $3x + 2y = 12$ and $4x - 5y = 10$, what is the value of $19x - 3y$?
Explanation:
Step 1
Solve the system of equations to find $x$ and $y$. Multiply the first equation by $5$ and the second by $2$: $\begin{cases} 15x + 10y = 60 \\ 8x - 10y = 20 \end{cases}$.
Step 2
Add the equations to eliminate $y$: $23x = 80$, wait, let's check numbers: $15x + 8x = 23x$, $60 + 20 = 80$, so $x = \frac{80}{23}$. Let's use a simpler linear combination method: notice we want $19x - 3y$. Can we combine the two equations directly? Let's check $a(3x + 2y) + b(4x - 5y) = 19x - 3y$.
Step 3
Equate coefficients: $3a + 4b = 19$ and $2a - 5b = -3$. Solving this gives $a = 3$ and $b = C$. Let's test $3(3x + 2y) + 2(4x - 5y) = 9x + 6y + 8x - 10y = 17x - 4y$. Let's re-verify the target expression or use direct elimination. Let's solve standard: $3x + 2y = 12 \implies y = 6 - \frac{3}{2}x$. Substitute into $4x - 5(6 - \frac{3}{2}x) = 10 \implies 4x - 30 + \frac{15}{2}x = 10 \implies \frac{23}{2}x = 40 \implies x = \frac{80}{23}$. Let's check if