🎯⭐ INTERACTIVE LESSON

Linear Equations and Inequalities

Learn step-by-step with interactive practice!

← Back to Standard Lesson

Linear Equations and Inequalities - Complete Interactive Lesson

Part 1: Linear Equations Basics

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.

Step	Work
Isolate $y$	$4y = -5x + 20$
Divide by 4	$y = -\frac{5}{4}x + 5$

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.

Step	Work
Identify slope	$12/month → $m = 12$
Identify y-intercept	$5 setup → $b = 5$
Write equation	$C = 12m + 5$

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.

Building Equations from Points 🎯

Identify the Form 🔍

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

Key Takeaways — Part 1

Form	Template	When to Use
Slope-intercept	$y = mx + b$	Know slope & y-intercept
Standard	$Ax + By = C$

Part 2: Multi-Step Equations

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

Part 3: Variables on Both Sides

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: $-3x + 6 > 12$ $-3x > 6$

Part 4: Systems of Equations

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

Part 5: Modeling with Equations

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

Identify the variables — what's changing? What's being measured?
Find the rate (slope) — the per-unit change
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? → → hours

Part 6: Problem-Solving Workshop

Linear Equations & Inequalities

Part 6 of 7 — Absolute Value and Literal Equations

Absolute Value Equations

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

$a x + b = c or a x +$

Part 7: Review & Applications

Linear Equations & Inequalities

Part 7 of 7 — SAT Mixed Practice & Review

Quick Reference

Concept	Formula/Rule
Slope-intercept	$y = mx + b$
Standard form	$Ax + By = C$ , slope

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

Step	Work
Point-slope setup	$y - 7 = 3(x - 2)$
Distribute	$y - 7 = 3x - 6$
Slope-intercept	$y = 3x + 1$

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

Step	Work
Find slope	$m = \frac{-8 - 4}{3 - (-1)} = \frac{-12}{4} = -3$
Point-slope	$y - 4 = -3(x + 1)$
Expand	$y = -3x - 3 + 4 = -3x + 1$
Standard form	$3x + y = 1$

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.

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

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 $x - 2y = -1$

Step	Work
Check for isolated variable	Second eq: $x = 2y - 1$ ✓ → use substitution
Substitute into first	$3(2y - 1) + 4y = 18$
Simplify	$6y - 3 + 4y = 18$ → $10y = 21$
Solve for $y$	$y = 2.1$
Back-substitute	$x = 2(2.1) - 1 = 3.2$

Worked Example 2 — Elimination with Multiplication

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

Step	Work
Make $x$ -coefficients match	Multiply eq 1 by 3, eq 2 by 2
New system	$6x + 15y = 3$ and $6x + 4y = -8$
Subtract	$11y = 11$ → $y = 1$
Back-substitute	$2x + 5(1) = 1$ → $x = -2$

Condition	Result	Lines
One solution	$x = a, y = b$	Lines intersect
No solution	$0 = k$ (contradiction)	Lines are parallel
Infinite solutions	$0 = 0$ (identity)	Lines are the same

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 $2x$ .

Step	Work
Subtract equations	$(4x + 3y) - (2x + 3y) = 17 - 11$

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

Worked Example 4

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

Step	Work
Add equations	$4x = 12$ → $x = 3$
Substitute	$3 + 2y = 5$ →

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

Strategy	When to Use
Substitution	One variable is already isolated
Elimination	Coefficients match or nearly match
Combo shortcut	SAT asks for an expression, not individual values

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

Compound Inequalities

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

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

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

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

Step	Work
Subtract 5	$-2x \geq 8$
Divide by $-2$ (FLIP!)	$x \leq -4$
Graph	Solid dot at $-4$ , shade left

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

Step	Work
Subtract 2 from all parts	$-9 < 3x \leq 12$
Divide all by 3	$-3 < x \leq 4$
Meaning	$x$ is between $-3$ (exclusive) and $4$ (inclusive)

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

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.

Constraint	Inequality
Total at least 60	$x + y \geq 60$
Math at most 40	$x \leq 40$
Both non-negative	$x \geq 0,\; y \geq 0$

Worked Example 4

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

Step	Work
Equation of line	$y = -2x + 4$
Dashed = strict	Use $<$ or $>$ (not $\leq$ or )

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

Rule	Detail
Flip when negative	Multiply/divide by a negative → reverse the inequality
Compound	Operate on all three parts simultaneously
Graphing: line type	Dashed = strict ( $<$ , $>$ ), Solid = inclusive ( $\leq$ , $\geq$ )
Graphing: shading	Above = $>$ or $\geq$ , Below = $<$ or $\leq$
System overlap	Solution is where both shaded regions intersect

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

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 $m$ :

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

Find the line parallel to $y = -4x + 9$ through the point $(2, 1)$ .

Step	Work
Same slope	$m = -4$
Point-slope	$y - 1 = -4(x - 2)$
Simplify	$y = -4x + 9$

Wait — same equation! This means $(2, 1)$ is actually ON the original line. Check: $1 = -4(2) + 9 = 1$ ✓

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

Step	Work
Original slope	$m = 3$
Perpendicular slope	$m_{\perp} = -1/3$
Point-slope	$y - 2 = -\frac{1}{3}(x - 6)$
Simplify	$y = -\frac{1}{3}x + 4$

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 $B(5, 7)$ .

Step	Work
Midpoint	$M = ((1+5)/2,\, (3+7)/2) = (3, 5)$
Slope of $\overline{AB}$	$m = (7-3)/(5-1) = 1$
Perpendicular slope	$m_{\perp} = -1$
Equation	$y - 5 = -1(x - 3)$ → $y = -x + 8$

Worked Example 4

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

Step	Work
Slope of first	$-6y = -3x + 12$ → $y = \frac{1}{2}x - 2$ →

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

Relationship	Slope Condition	Example
Parallel	$m_1 = m_2$	$y = 3x + 1$ ∥ $y = 3x - 5$
Perpendicular	$m_1 \cdot m_2 = -1$	$y = 2x$ ⊥
Neither	Slopes differ but product $\neq -1$	$y = 2x$ and $y = 3x$

Point-slope form is your friend: $y - y_1 = m(x - x_1)$
Always convert standard form to slope-intercept before comparing slopes

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

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?

Step	Work
Company A cost	$C_A = 0.05t + 40$
Company B cost	$C_B = 0.15t + 25$
Set equal	$0.05t + 40 = 0.15t + 25$
Solve	$15 = 0.10t$ → $t = 150$ texts
Verify	$C_A = 0.05(150) + 40 = 47.50$ ✓

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.

Component	Value	Real-World Meaning
Slope	$0.12$	Each additional minute costs $0.12
Y-intercept	$35$	The base cost with zero minutes is $35
$C(100)$	$47$	Using 100 minutes costs $47

Worked Example 3

From a table:

Hours Worked ( $x$ )	Pay ( $y$ )
$0$	$50$
$4$	$110$

Step	Work
Slope	$(110 - 50)/(4 - 0) = 60/4 = 15$
Y-intercept	$50$ (from the table directly)

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

Concept	How to Find It
Slope from context	Rate per unit ($/hour, miles/gallon, etc.)
Y-intercept from context	Starting value, initial amount, base cost
Slope from table	$\Delta y / \Delta x$ using any two rows
"When are they equal?"	Set the two expressions equal

"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

a x + b = c or a x + 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)

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

Step	Work
Case 1	$4x + 1 = 11$ → $4x = 10$ → $x = 5/2$
Case 2	$4x + 1 = -11$ → $4x = -12$ → $x = -3$
Solutions	$x = 5/2$ or $x = -3$
Verify	$

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

Step	Work
Set up compound	$-3 \leq x - 5 \leq 3$
Add 5	$2 \leq x \leq 8$
In interval form	$[2, 8]$

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 $h$ .

Step	Work
Multiply by 2	$2A = bh$
Divide by $b$	$h = \frac{2A}{b}$

Worked Example 4

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

Step	Work
Common denominator	$\frac{1}{R} = \frac{R_2 + R_1}{R_1 R_2}$

Worked Example 5

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

Step	Work
Subtract 32	$F - 32 = \frac{9}{5}C$
Multiply by $\frac{5}{9}$

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

Absolute Value Scenario	Setup	Solution Type
$	expr	= c$ (c > 0)
$	expr	= 0$
$	expr	= -c$ (c > 0)
$	expr	< c$
$	expr	> c$

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

Common SAT Mistakes to Avoid

Forgetting to flip the inequality sign when dividing by a negative
Misreading what the question asks — "What is $x + y$ ?" vs "What is $x$ ?"
Not checking answer choices by plugging back in
Rushing standard form → slope conversion (slope is $-A/B$ , not $A/B$ )
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 $k$ .

Step	Work
Find x-intercept of $g$	$0 = -x + 8$ → $x = 8$ → point $(8$

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 $\ell$ ?

Step	Work
Slope of given line	$y = -\frac{5}{2}x + 5$ → $m = -5/2$

Worked Example 3

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

Step	Work
Isolate absolute value	$
Case 1	$2x - 1 = 7$ → $x = 4$
Case 2	$2x - 1 = -7$ →

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)

Topic	Core Skill	Common Trap
Forms of lines	Convert fluently	Slope from standard form is $-A/B$
Systems	Choose sub vs. elim	Read what they ask ( $x$ ? $y$ ? $x+y$ ?)
Inequalities	Flip on × or ÷ by negative	"At most" = $\leq$ , "at least" = $\geq$
Parallel/Perp	Compare slopes	Perpendicular: $m_1 \cdot m_2 = -1$
Word problems	Slope = rate, $b$ = start	Decreasing = negative slope
Absolute value	Split into two cases	$
Literal equations	Isolate target variable	Treat other letters as numbers

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

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 and Inequalities

Linear Equations and Inequalities - Complete Interactive Lesson

Part 1: Linear Equations Basics

Linear Equations & Inequalities

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

Standard Form: Ax+By=CAx + By = CAx+By=C

Worked Example 1

Worked Example 2

SAT Trap ⚠️

Point-Slope Form: y−y1=m(x−x1)y - y_1 = m(x - x_1)y−y1​=m(x−x1​)

Key Takeaways — Part 1

Part 2: Multi-Step Equations

Linear Equations & Inequalities

Method 1: Substitution

Part 3: Variables on Both Sides

Linear Equations & Inequalities

Solving Linear Inequalities

Part 4: Systems of Equations

Linear Equations & Inequalities

Parallel Lines

Perpendicular Lines

Part 5: Modeling with Equations

Linear Equations & Inequalities

Setting Up Linear Models

Common SAT Word Problem Types

Part 6: Problem-Solving Workshop

Linear Equations & Inequalities

Absolute Value Equations

Part 7: Review & Applications

Linear Equations & Inequalities

Quick Reference

Worked Example 3

Worked Example 4

Slope Formula Shortcut

Method 2: Elimination

Worked Example 1 — Deciding Which Method

Worked Example 2 — Elimination with Multiplication

Special Cases

SAT Strategy 💡

The "Combo" Shortcut

Worked Example 3

Worked Example 4

Key Takeaways — Part 2

Compound Inequalities

Worked Example 1

Worked Example 2

Graphing Inequalities

SAT Pattern ⚠️

Systems of Inequalities on the SAT

Worked Example 3

Worked Example 4

Key Takeaways — Part 3

Finding the Equation of a Line

Worked Example 1

Worked Example 2

Midpoint and Distance

Perpendicular Bisectors

Worked Example 3

Worked Example 4

Key Takeaways — Part 4

Worked Example 1

Reading Tables on the SAT

Interpreting Slope and Y-Intercept in Context

Worked Example 2

Worked Example 3

Key Takeaways — Part 5

Absolute Value Inequalities

Worked Example 1

Worked Example 2

Literal Equations (Solving for a Variable)

Worked Example 3

Worked Example 4

Worked Example 5

Key Takeaways — Part 6

Common SAT Mistakes to Avoid

Time-Saving Strategies

SAT-Style Hard Problems: Worked Solutions

Worked Example 1

Worked Example 2

Worked Example 3

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

Standard Form: $Ax + By = C$

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