Linear Equations and Inequalities

Linear Equations & Inequalities

Part 1 of 7 — Slope-Intercept and Standard Form

The SAT Math section heavily tests your ability to work with linear equations. You'll see these in both calculator and no-calculator modules. Mastering the two main forms — and converting between them — is essential.

Slope-Intercept Form: $y = mx + b$

• $m$ = slope (rate of change)
• $b$ = y-intercept (value when $x = 0$)

Example: A phone plan charges $45/month plus $0.10 per text. If $y$ is the monthly cost and $x$ is the number of texts:

$y = 0.10x + 45$

Standard Form: $Ax + By = C$

• Useful for finding intercepts quickly
• x-intercept: set $y = 0$ → $x = C/A$
• y-intercept: set $x = 0$ → $y = C/B$

Worked Example 1

Convert $5x + 4y = 20$ to slope-intercept form and identify the intercepts.

Worked Example 2

A streaming service costs $12/month after a one-time $5 setup fee. Write the total cost $C$ after $m$ months and find the cost after 6 months.

SAT Trap ⚠️

When the SAT gives you standard form and asks for the slope, students often forget to isolate $y$ first. The slope is NOT just $A/B$ — it's $-A/B$.

Slope-Intercept & Standard Form 🎯

Point-Slope Form: $y - y_1 = m(x - x_1)$

This form is ideal when you know a point on the line and the slope.

Worked Example 3

Write the equation of a line through $(2, 7)$ with slope $3$.

Worked Example 4

A line passes through $(-1, 4)$ and $(3, -8)$. Find its equation in standard form.

Slope Formula Shortcut

$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\text{rise}}{\text{run}}$

SAT Tip: When two points are given and the question asks for the equation, always compute the slope first.

Building Equations from Points 🎯

Identify the Form 🔍

For each equation, select the correct form it is written in.

Key Takeaways — Part 1

• Real-world problems: the rate = slope, the starting value = y-intercept
• Always isolate $y$ before identifying the slope from standard form
• The slope formula $m = \frac{y_2 - y_1}{x_2 - x_1}$ works with any two points on the line
• On the SAT, check which form the answer choices use before you start solving

Linear Equations & Inequalities

Part 2 of 7 — Systems of Linear Equations

Systems of equations appear on nearly every SAT. You need to be fast and flexible with solving methods.

Method 1: Substitution

Best when one variable is already isolated.

Example: $y = 2x + 1$ $3x + y = 11$

Substitute: $3x + (2x + 1) = 11$ → $5x = 10$ → $x = 2$, $y = 5$

Method 2: Elimination

Best when coefficients can be matched easily.

Example: $2x + 3y = 7$ $2x - y = 3$

Subtract: $4y = 4$ → $y = 1$, $x = 2$

Worked Example 1 — Deciding Which Method

Solve: $3x + 4y = 18$ and $x - 2y = -1$

Worked Example 2 — Elimination with Multiplication

Solve: $2x + 5y = 1$ and $3x + 2y = -4$

Special Cases

SAT Strategy 💡

If the SAT asks "For what value of $k$ does the system have no solution?" — make the slopes equal but the y-intercepts different. Parallel lines = no solution.

Systems of Equations — Solving 🎯

The "Combo" Shortcut

Sometimes the SAT asks for an expression like $x + y$ or $2x - y$ rather than individual values. You can often find these directly!

Worked Example 3

Given: $4x + 3y = 17$ and $2x + 3y = 11$. Find $2x$.

No need to find $x$ and $y$ separately!

Worked Example 4

Given: $x + 2y = 5$ and $3x - 2y = 7$. Find $x + y$.

SAT Tip: Always check if the requested expression can be obtained by adding or subtracting the two equations before solving individually.

Systems — Harder Problems 🎯

Classify the System 🔍

For each system, determine the number of solutions.

Key Takeaways — Part 2

• No solution: same slope, different intercept (parallel lines)
• Infinite solutions: same slope AND same intercept (same line)
• One solution: different slopes (lines intersect)
• Always read what the question asks — $x$? $y$? $x + y$? $3x - 2y$?
• If the answer choices are simple numbers, try back-solving

Linear Equations & Inequalities

Part 3 of 7 — Linear Inequalities

The SAT tests inequalities in both algebraic and graphical form.

Solving Linear Inequalities

Same rules as equations EXCEPT: flip the inequality sign when multiplying or dividing by a negative.

Example: $-3x + 6 > 12$ $-3x > 6$ $x < -2 \quad \text{(flip!)}$

Compound Inequalities

$-1 < 2x + 3 \leq 9$

Subtract 3 from all parts: $-4 < 2x \leq 6$

Divide by 2: $-2 < x \leq 3$

Worked Example 1

Solve $5 - 2x \geq 13$ and graph the solution.

Worked Example 2

Solve the compound inequality $-7 < 3x + 2 \leq 14$.

Graphing Inequalities

• $y > mx + b$: shade above the line, dashed boundary
• $y \leq mx + b$: shade below the line, solid boundary
• The solution to a system of inequalities is the overlap region

SAT Pattern ⚠️

The SAT loves: "Which point is in the solution set of $y > 2x - 1$ and $y < -x + 5$?" Plug each answer choice into BOTH inequalities and check.

Inequalities — Basics 🎯

Systems of Inequalities on the SAT

When two inequalities define a region, the SAT typically asks:

• "Which point is in the solution region?"
• "Which inequality represents the shaded region?"

Worked Example 3

A student needs at least 60 hours of study across two subjects. They spend $x$ hours on math and $y$ hours on science, with at most 40 hours on math. Write the system.

Worked Example 4

From a graph: a dashed line through $(0, 4)$ with slope $-2$, shaded below. Write the inequality.

SAT Tip: Solid line = $\leq$ or $\geq$. Dashed line = $<$ or $>$. Always check the line type before selecting!

Inequality Applications 🎯

Inequality Symbols 🔍

Match each phrase to the correct inequality symbol.

Key Takeaways — Part 3

• To check a point: plug into BOTH inequalities — both must be true
• "At most" = $\leq$, "at least" = $\geq$, "more than" = $>$, "fewer than" = $<$
• Compound inequalities preserve $\leq$ vs $<$ when dividing by a positive

Linear Equations & Inequalities

Part 4 of 7 — Parallel and Perpendicular Lines

These concepts appear frequently in SAT geometry-meets-algebra questions.

Parallel Lines

• Same slope, different y-intercepts
• $y = 3x + 2$ is parallel to $y = 3x - 5$

Perpendicular Lines

• Slopes are negative reciprocals: $m_1 \cdot m_2 = -1$
• $y = 2x + 1$ is perpendicular to $y = -\frac{1}{2}x + 4$

Finding the Equation of a Line

Given a point $(x_1, y_1)$ and slope $m$:

$y - y_1 = m(x - x_1) \quad \text{(point-slope form)}$

Worked Example 1

Find the line parallel to $y = -4x + 9$ through the point $(2, 5)$.

Worked Example 2

Find the line perpendicular to $y = 3x + 1$ passing through $(6, 2)$.

Midpoint and Distance

• Midpoint: $\left(\frac{x_1 + x_2}{2},\, \frac{y_1 + y_2}{2}\right)$
• Distance: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Parallel & Perpendicular — Basics 🎯

Perpendicular Bisectors

A perpendicular bisector of a segment passes through its midpoint at a right angle. This combines midpoint, perpendicular slope, and point-slope concepts.

Worked Example 3

Find the perpendicular bisector of the segment from $A(1, 3)$ to $B(5, 7)$.

Worked Example 4

The line $3x - 6y = 12$ is parallel to $kx + 4y = 8$. Find $k$.

SAT Tip: When comparing slopes from standard form, convert BOTH to slope-intercept. Don't try to compare standard form coefficients directly.

Parallel & Perpendicular — Harder Problems 🎯

Classify Line Relationships 🔍

For each pair of lines, determine their relationship.

Key Takeaways — Part 4

• Point-slope form is your friend: $y - y_1 = m(x - x_1)$
• Always convert standard form to slope-intercept before comparing slopes
• Midpoint $= \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$
• Distance $= \sqrt{(\Delta x)^2 + (\Delta y)^2}$

Linear Equations & Inequalities

Part 5 of 7 — Word Problems with Linear Models

The SAT tests whether you can translate real-world scenarios into linear equations.

Setting Up Linear Models

1. Identify the variables — what's changing? What's being measured?
2. Find the rate (slope) — the per-unit change
3. Find the starting value (y-intercept) — the initial amount

Common SAT Word Problem Types

Type 1 — Cost/Revenue: A rideshare charges $3 base + $1.50/mile. Total cost for $m$ miles: $C = 1.50m + 3$

Type 2 — Distance/Rate/Time: Two trains leave at the same time. Train A: 60 mph. Train B: 80 mph. If B starts 30 miles behind, when does B catch A? $80t = 60t + 30$ → $20t = 30$ → $t = 1.5$ hours

Type 3 — "Already...and then...": A pool has 200 gallons and is being filled at 15 gallons/minute. After $t$ minutes: $V = 15t + 200$

Worked Example 1

A cellphone company charges $40/month for a plan plus $0.05 per text message. Another company charges $25/month plus $0.15 per text. How many texts make the costs equal?

Reading Tables on the SAT

When given a table, calculate slope: $m = \frac{\Delta y}{\Delta x}$ using any two rows. Then find $b$ by plugging in one point.

Word Problems — Setup 🎯

Interpreting Slope and Y-Intercept in Context

The SAT frequently asks questions like:

• "What does the slope represent in this context?"
• "What is the meaning of the y-intercept?"

Worked Example 2

The equation $C = 0.12m + 35$ models a monthly phone bill, where $C$ is the cost in dollars and $m$ is the number of minutes used.

Worked Example 3

From a table:

SAT Tip: The SAT may phrase slope interpretation as "For every increase of 1 in $x$, $y$ increases/decreases by ___." The answer is the slope.

Interpreting Models 🎯

Identify the Slope 🔍

For each scenario, select the correct slope value.

Key Takeaways — Part 5

• "Draining/decreasing" = negative slope; "filling/increasing" = positive slope
• From a table: slope $= \Delta y / \Delta x$, then plug in a point for $b$
• Slope interpretation: "For every 1-unit increase in $x$, $y$ changes by $m$"
• Check your model: plug a known data point back in to verify

Linear Equations & Inequalities

Part 6 of 7 — Absolute Value and Literal Equations

Absolute Value Equations

$|ax + b| = c$ splits into two cases (when $c \geq 0$):

$ax + b = c \quad \text{or} \quad ax + b = -c$

Example: $|2x - 3| = 7$

• Case 1: $2x - 3 = 7$ → $x = 5$
• Case 2: $2x - 3 = -7$ → $x = -2$

⚠️ If $|ax + b| = -k$ where $k > 0$: no solution (absolute value is never negative).

Absolute Value Inequalities

• $|x| < a$: $-a < x < a$ (AND — between)
• $|x| > a$: $x < -a$ or $x > a$ (OR — outside)

Worked Example 1

Solve $|4x + 1| = 11$.

Worked Example 2

Solve $|x - 5| \leq 3$.

Absolute Value 🎯

Literal Equations (Solving for a Variable)

The SAT often asks you to rearrange a formula. Treat every other variable as a number.

Worked Example 3

Solve $A = \frac{1}{2}bh$ for $h$.

Worked Example 4

Solve $\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2}$ for $R$.

Worked Example 5

Solve $F = \frac{9}{5}C + 32$ for $C$.

SAT Strategy 💡: For literal equations, treat every other variable as a number. The solving process is identical — just letters instead of digits.

Literal Equations 🎯

Absolute Value Solutions 🔍

For each equation/inequality, select the number of solutions.

Key Takeaways — Part 6

• Literal equations: isolate the target variable using normal algebra steps
• Treat all other variables as constants when solving for one variable
• Always verify absolute value solutions by plugging back in

Linear Equations & Inequalities

Part 7 of 7 — SAT Mixed Practice & Review

Quick Reference

Common SAT Mistakes to Avoid

1. Forgetting to flip the inequality sign when dividing by a negative
2. Misreading what the question asks — "What is $x + y$?" vs "What is $x$?"
3. Not checking answer choices by plugging back in
4. Rushing standard form → slope conversion (slope is $-A/B$, not $A/B$)
5. Absolute value = negative → instant "no solution"

Time-Saving Strategies

• Back-solve from answer choices — plug in each option when algebra is messy
• Pick smart numbers — if the problem has fractions, choose a common denominator
• Look for shortcuts — many system problems can be solved by adding/subtracting the equations directly
• Estimate first — eliminate obviously wrong answers before computing

Mixed Review — Round 1 🎯

SAT-Style Hard Problems: Worked Solutions

Worked Example 1

The function $f(x) = 3x + k$ passes through the point where $g(x) = -x + 8$ crosses the x-axis. Find $k$.

Worked Example 2

In the $xy$-plane, line $\ell$ passes through the origin and is perpendicular to $5x + 2y = 10$. Which point is on line $\ell$?

Worked Example 3

If $|2x - 1| + 3 = 10$, what is the product of the possible values of $x$?

Mixed Review — Round 2 (Hard) 🎯

Speed Round: What Strategy? 🔍

For each problem type, select the fastest solving approach.

Key Takeaways — Part 7 (Full Topic Review)

Final SAT Strategies:

• Back-solve from answer choices is your best friend on hard problems
• Estimate to eliminate obviously wrong answers
• Have your approach ready in the first 5 seconds — don't stare