🎯⭐ INTERACTIVE LESSON

Linear Equations and Inequalities

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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.

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 $x = 0$ → $y = C/B$

Converting Between Forms

To convert $3x + 2y = 12$ to slope-intercept: $2y = -3x + 12$ $y = -\frac{3}{2}x + 6$

So slope $= -3/2$ and y-intercept $= 6$ .

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 🎯

Key Takeaways — Part 1

Slope-intercept ( $y = mx + b$ ): slope is the coefficient of $x$ , y-int is the constant
Standard form ( $Ax + By = C$ ): slope , NOT

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 (same direction, B is behind). When does B catch A?

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

Real-world problems: the rate = slope, the starting value = y-intercept

Always isolate

y

before identifying the slope from standard form

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: $4y = 4$ → $y = 1$ , $x = 2$

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 🎯

Key Takeaways — Part 2

Substitution: best when a variable is isolated ( $y = ...$ )
Elimination: best when coefficients match or nearly match
No solution: same slope, different intercept (parallel lines)
Infinite solutions: same slope AND same intercept (same line)
SAT shortcut: to find a combo like $x + y$ , look for ways to avoid solving for individual variables

Compound Inequalities

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

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

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

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 🎯

Key Takeaways — Part 3

Flip the inequality when multiplying/dividing by a negative
Compound inequalities: perform the same operation on all three parts
Graphing: $>$ or $<$ = dashed line; $\geq$ or $\leq$ = solid line
To check a point: plug into both inequalities — both must be true

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 $m$ :

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

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

Perpendicular slope: $m = -1/3$
$y - 2 = -\frac{1}{3}(x - 6)$
$y = -\frac{1}{3}x + 4$

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 Lines 🎯

Key Takeaways — Part 4

Parallel: same slope ( $m_1 = m_2$ )
Perpendicular: negative reciprocal slopes ( $m_1 \cdot m_2 = -1$ )
Point-slope form is your friend: $y - y_1 = m(x - x_1)$
Always convert standard form to slope-intercept before comparing slopes

Distance A = $60t$ , Distance B = $80t$
B catches A when $80t = 60t + d$ (where $d$ is the head start)

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

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 🎯

Key Takeaways — Part 5

Identify slope (rate) and y-intercept (starting value) from word problems
"Draining/decreasing" = negative slope; "filling/increasing" = positive slope
From a table: slope $= \Delta y / \Delta x$ , then plug in a point for $b$
"When are they equal?" → set the two expressions equal to each other

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)

Literal Equations (Solving for a Variable)

The SAT often asks you to rearrange a formula.

Example: Solve $A = \frac{1}{2}bh$ for $h$ : $h = \frac{2A}{b}$

Example: Solve $F = \frac{9}{5}C + 32$ for $C$ : $C = \frac{5(F - 32)}{9}$

For literal equations, treat every other variable as a number and solve normally. The algebra is the same — just letters instead of digits.

Absolute Value & Literal Equations 🎯

Key Takeaways — Part 6

$|expr| = c$ → two cases: $expr = c$ or $expr = -c$
$|expr| = \text{negative}$ → no solution
$|expr| < c$ → compound inequality (between): $-c < expr < c$
$|expr| > c$ → two separate inequalities (outside): $expr < -c$ or $expr > c$
Literal equations: isolate the target variable using normal algebra steps

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 $A/B$ )

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

Mixed Review 🎯

Key Takeaways — Part 7

Master all forms of linear equations and when to use each
Systems: elimination is usually faster than substitution on the SAT
Always check what the question is asking for — $x$ ? $y$ ? $x + y$ ? $5x$ ?
Back-solving from answer choices is a powerful SAT-specific strategy
Practice converting between forms fluently — speed matters on test day

Linear Equations and Inequalities

Method 2: Elimination

Special Cases

SAT Strategy 💡

Key Takeaways — Part 2

Compound Inequalities

Graphing Inequalities

SAT Pattern ⚠️

Key Takeaways — Part 3

Finding the Equation of a Line

Midpoint and Distance

Key Takeaways — Part 4

Reading Tables on the SAT

Key Takeaways — Part 5

Absolute Value Inequalities

Literal Equations (Solving for a Variable)

SAT Strategy 💡

Key Takeaways — Part 6

Common SAT Mistakes to Avoid

Time-Saving Strategies

Key Takeaways — Part 7

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

Converting Between Forms

SAT Trap ⚠️

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

Method 2: Elimination

Special Cases

SAT Strategy 💡

Key Takeaways — Part 2

Compound Inequalities

Graphing Inequalities

SAT Pattern ⚠️

Key Takeaways — Part 3

Finding the Equation of a Line

Midpoint and Distance

Key Takeaways — Part 4

Reading Tables on the SAT

Key Takeaways — Part 5

Absolute Value Inequalities

Literal Equations (Solving for a Variable)

SAT Strategy 💡

Key Takeaways — Part 6

Common SAT Mistakes to Avoid

Time-Saving Strategies

Key Takeaways — Part 7

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

Standard Form: $Ax + By = C$