Comprehensive Guide to Linear Algebra for the SAT

Mastering Linear Equations, Functions, Systems, and Inequalities

This comprehensive guide covers all essential algebra topics tested on the SAT math section, which comprises approximately 35% of the exam. Each section provides detailed explanations, step-by-step solution methods, and complete worked examples to help you master linear algebra concepts.

1. Linear Equations in One Variable

Linear equations in one variable focus on finding the value of a single variable (usually x). While they may seem simple, these equations can become complex with fractions, parentheses, and multiple steps.

The 4-Step Process

Step 1: Simplify Each Side of the Equation

Combine like terms and eliminate parentheses to make the equation manageable.

Example 1: Combining Like Terms

5x + 3 - 2x + 4 = 10
3x + 7 = 10
```

**Example 2: Eliminating Parentheses**
```
2(3x + 4) - 5 = 11
6x + 8 - 5 = 11
6x + 3 = 11
```

**Example 3: Simplifying Complex Fractions**
```
x/4 + 3/6 = 2/3
Multiply all terms by 12 (common denominator):
3x + 6 = 8
```

**Example 4: Complete Simplification**
```
3(x + 2) - 2x = 5 + x - 1
3x + 6 - 2x = 5 + x - 1
3x - 2x - x = 5 - 1 - 6
-x = -2
```

**Step 2: Get the Variable on One Side**

Isolate the variable using inverse operations.

**Example 5: Undoing Addition**
```
x + 4 = 9
x = 9 - 4
x = 5
```

**Example 6: Undoing Multiplication**
```
4x = 20
x = 20/4
x = 5
```

**Example 7: Combined Operations**
```
3x + 7 = 10
3x = 10 - 7
3x = 3
x = 1
```

**Step 3: Simplify the Equation Again**

Ensure your solution is in simplest form.

**Example 8: Combining Variables**
```
2x + 3x = 10
5x = 10
x = 2
```

**Example 9: Reducing Fractions**
```
8/4 = x
2 = x
```

**Example 10: Converting Decimals**
```
x = 0.75
x = 3/4 or x = 75%
```

**Step 4: Check Your Solution**

Always verify by substituting back into the original equation.

**Example 11: Verification**
```
Original: 3x + 4 = 13, Solution: x = 3
Check: 3(3) + 4 = 9 + 4 = 13 ✓
```

### Additional Practice Examples

**Example 12: Multi-Step Equation with Fractions**
```
(2x + 6)/3 = 4
2x + 6 = 12
2x = 6
x = 3
```

**Example 13: Equation with Variables on Both Sides**
```
5x - 3 = 2x + 9
5x - 2x = 9 + 3
3x = 12
x = 4
```

**Example 14: Complex Parentheses**
```
4(2x - 1) + 3(x + 2) = 25
8x - 4 + 3x + 6 = 25
11x + 2 = 25
11x = 23
x = 23/11
```

**Example 15: Nested Fractions**
```
x/2 + x/3 = 10
Multiply by 6: 3x + 2x = 60
5x = 60
x = 12
```

---

## 2. Linear Equations in Two Variables

These equations involve two variables (typically x and y) and can be expressed in different forms.

### Standard Form vs. Slope-Intercept Form

**Standard Form:** ax + by = c
**Slope-Intercept Form:** y = mx + b (where m = slope, b = y-intercept)

### Conversion Examples

**Example 1: Standard to Slope-Intercept**
```
2x + 3y = 6
3y = -2x + 6
y = -2/3x + 2
```

**Example 2: Slope-Intercept to Standard**
```
y = -2/3x + 2
Multiply by 3: 3y = -2x + 6
Rearrange: 2x + 3y = 6
```

**Example 3: Converting with Larger Coefficients**
```
4x + 6y = 12
6y = -4x + 12
y = -2/3x + 2
```

### Understanding the Graph

**Example 4: Positive Slope**
```
y = 2x + 3
Slope: 2 (rises 2 units for every 1 unit right)
Y-intercept: (0, 3)
```

**Example 5: Negative Slope**
```
y = -x - 2
Slope: -1 (falls 1 unit for every 1 unit right)
Y-intercept: (0, -2)
```

### Solving Methods

**Example 6: Substitution Method**
```
System: y = 2x + 3 and y = -x - 2
Set equal: 2x + 3 = -x - 2
3x = -5
x = -5/3
Substitute: y = 2(-5/3) + 3 = -10/3 + 9/3 = -1/3
Solution: x = -5/3, y = -1/3
```

**Example 7: Elimination Method**
```
System: 2x + 3y = 6 and x - y = 2
Multiply second by 3: 3x - 3y = 6
Add equations: 5x = 12
x = 12/5
Substitute: 12/5 - y = 2
y = 2/5
```

**Example 8: Graphical Method**
```
y = -x + 1 and y = 1/2x - 1
Graph both lines and find intersection
Solution: x = 4/3, y = -1/3
```

### Word Problem Examples

**Example 9: Cost Problem**
```
Scenario: Total cost with $20 base + $10 per GB
Equation: C(d) = 10d + 20
If d = 5: C(5) = 10(5) + 20 = $70
```

**Example 10: Item Pricing**
```
3x + 5y = 11 (x = apples, y = oranges)
If x = 2: 3(2) + 5y = 11
6 + 5y = 11
5y = 5
y = 1
2 apples and 1 orange cost $11
```

**Example 11: Rate Comparison**
```
Runner A: D_A(t) = 6t + 2
Runner B: D_B(t) = 8t
When do they meet?
6t + 2 = 8t
2 = 2t
t = 1 hour
```

### Special Cases

**Example 12: No Solution**
```
2x + 3y = 6 and 2x + 3y = 10
Subtract: 0 = 4 (contradiction)
No solution (parallel lines)
```

**Example 13: Infinitely Many Solutions**
```
2x + 3y = 6 and 4x + 6y = 12
Second equation is 2× the first
Infinitely many solutions (same line)
```

**Example 14: Finding Coefficients**
```
If ax + 3y = 12 passes through (2, 2):
a(2) + 3(2) = 12
2a + 6 = 12
2a = 6
a = 3
```

**Example 15: Perpendicular Lines**
```
Line 1: y = 2x + 3 (slope = 2)
Perpendicular line: y = -1/2x + b (slope = -1/2)
If passes through (0, 5): b = 5
Equation: y = -1/2x + 5
```

---

## 3. Linear Functions

A linear function creates a straight-line graph and is expressed as f(x) = mx + b.

### Basic Function Concepts

**Example 1: Simple Linear Function**
```
f(x) = x + 2
If x = 3: f(3) = 3 + 2 = 5
If x = -1: f(-1) = -1 + 2 = 1
```

**Example 2: Function with Slope**
```
f(x) = 2x + 3
Slope = 2, Y-intercept = 3
If x = 0: f(0) = 3 (starting point)
If x = 1: f(1) = 5 (increases by 2)
```

**Example 3: Negative Slope Function**
```
f(x) = -3x + 6
Slope = -3 (decreases 3 units per unit increase in x)
If x = 0: f(0) = 6
If x = 2: f(2) = 0
```

### Converting Between Forms

**Example 4: Standard to Function Form**
```
3x + 4y = 12
4y = -3x + 12
y = -3/4x + 3
f(x) = -3/4x + 3
```

**Example 5: Function to Standard Form**
```
f(x) = -2/3x + 2
y = -2/3x + 2
Multiply by 3: 3y = -2x + 6
Standard form: 2x + 3y = 6
```

### Graphing Linear Functions

**Example 6: Plotting with Slope and Intercept**
```
f(x) = 1/2x + 1
Start at (0, 1)
Rise 1, run 2: next point (2, 2)
Rise 1, run 2: next point (4, 3)
```

**Example 7: Graphing Negative Slope**
```
f(x) = -2x + 4
Start at (0, 4)
Fall 2, run 1: next point (1, 2)
Fall 2, run 1: next point (2, 0)
```

**Example 8: Identifying Function from Graph**
```
Graph passes through (0, 2) and (3, 8)
Slope = (8-2)/(3-0) = 6/3 = 2
Y-intercept = 2
Function: f(x) = 2x + 2
```

### Word Problem Applications

**Example 9: Phone Plan Cost**
```
Base cost: $20, Data: $10/GB
C(d) = 10d + 20
For 5 GB: C(5) = 10(5) + 20 = $70
For 8 GB: C(8) = 10(8) + 20 = $100
```

**Example 10: Runner Distance**
```
Runner A (head start): D_A(t) = 6t + 2
Runner B (faster): D_B(t) = 8t
When does B catch A?
6t + 2 = 8t
t = 1 hour
```

**Example 11: Temperature Conversion**
```
F(C) = 9/5C + 32
Convert 20°C to Fahrenheit:
F(20) = 9/5(20) + 32 = 36 + 32 = 68°F
```

**Example 12: Rental Car Pricing**
```
Base: $30, Per mile: $0.25
C(m) = 0.25m + 30
For 100 miles: C(100) = 0.25(100) + 30 = $55
```

### Comparing Functions

**Example 13: Steeper Slope**
```
f(x) = 2x + 1 vs g(x) = 5x + 1
g(x) has steeper slope (5 > 2)
Both start at y = 1
```

**Example 14: Different Intercepts**
```
f(x) = 2x + 3 vs g(x) = 2x - 1
Same slope (parallel lines)
f(x) is 4 units above g(x)
```

**Example 15: Finding Intersection**
```
f(x) = 3x + 2 and g(x) = -x + 10
Set equal: 3x + 2 = -x + 10
4x = 8
x = 2, y = 8
Intersection: (2, 8)
```

---

## 4. Systems of Two Linear Equations in Two Variables

Systems involve finding values that satisfy multiple equations simultaneously.

### Substitution Method

**Example 1: Basic Substitution**
```
2y + 3x = 14
y - 2x = 0

From second: y = 2x
Substitute: 2(2x) + 3x = 14
4x + 3x = 14
7x = 14
x = 2, y = 4
```

**Example 2: Substitution with Fractions**
```
4x + y = 22
2x - y = 6

From second: y = 2x - 6
Substitute: 4x + (2x - 6) = 22
6x = 28
x = 14/3, y = 10/3
```

**Example 3: More Complex Substitution**
```
3x + 4y = 12
x - 2y = 3

From second: x = 2y + 3
Substitute: 3(2y + 3) + 4y = 12
6y + 9 + 4y = 12
10y = 3
y = 3/10, x = 18/5
```

### Elimination Method

**Example 4: Direct Elimination**
```
2y + 4x = 20
y - x = 1

Multiply second by 2: 2y - 2x = 2
Subtract: 6x = 18
x = 3, y = 4
```

**Example 5: Elimination with Multiplication**
```
6x + 5y = 7
3x - 2y = 4

Multiply second by 2: 6x - 4y = 8
Subtract: 9y = -1
y = -1/9, x = 34/27
```

**Example 6: Adding to Eliminate**
```
5x - 3y = 2
2x + y = 9

Multiply second by 3: 6x + 3y = 27
Add: 11x = 29
x = 29/11, y = 41/11
```

**Example 7: Aligning Coefficients**
```
3x + 2y = 8
5x - 2y = 12

Add directly (y cancels): 8x = 20
x = 5/2, y = 5/4
```

### Graphing Method

**Example 8: Finding Intersection Graphically**
```
2y + 4x = 20 → y = -2x + 10
y - x = 1 → y = x + 1

Plot both lines
Intersection at (3, 4)
```

**Example 9: Slope-Intercept Graphing**
```
y = -x + 1
y = 1/2x - 1

Graph shows intersection at (4/3, -1/3)
```

**Example 10: Parallel Lines (No Solution)**
```
y = 2x + 3
y = 2x - 1

Same slope, different intercepts
Lines never intersect
No solution
```

### Complex Examples

**Example 11: Decimal Coefficients**
```
0.5x + 0.3y = 2.3
0.2x - 0.1y = 0.3

Multiply first by 10: 5x + 3y = 23
Multiply second by 10: 2x - y = 3
Solve: x = 2, y = 13/3
```

**Example 12: Three-Step Elimination**
```
4x + 3y = 11
5x - 2y = 8

Multiply first by 2: 8x + 6y = 22
Multiply second by 3: 15x - 6y = 24
Add: 23x = 46
x = 2, y = 1
```

**Example 13: Substitution with Distribution**
```
2(x + y) = 10
3x - y = 5

From first: x + y = 5, so y = 5 - x
Substitute: 3x - (5 - x) = 5
4x = 10
x = 5/2, y = 5/2
```

**Example 14: Word Problem System**
```
Total tickets sold: x + y = 100
Total revenue: 5x + 3y = 420

From first: y = 100 - x
Substitute: 5x + 3(100 - x) = 420
2x = 120
x = 60 adults, y = 40 children
```

**Example 15: Mixture Problem**
```
Volume: x + y = 10 (liters)
Concentration: 0.2x + 0.5y = 3.5

From first: x = 10 - y
Substitute: 0.2(10 - y) + 0.5y = 3.5
0.3y = 1.5
y = 5, x = 5
```

---

## 5. Linear Inequalities in One or Two Variables

Inequalities use <, ≤, >, ≥ instead of = and require careful attention to inequality direction.

### Basic Inequalities (No Reversal)

**Example 1: Simple Inequality**
```
2x - 3 < 7
2x < 10
x < 5
```

**Example 2: Multi-Step Inequality**
```
3x + 5 ≤ 14
3x ≤ 9
x ≤ 3
```

**Example 3: With Fractions**
```
x/2 + 3 > 7
x/2 > 4
x > 8
```

### Inequalities Requiring Reversal

**Example 4: Negative Coefficient**
```
-3x + 4 > 1
-3x > -3
x < 1 (reversed!)
```

**Example 5: Division by Negative**
```
-4x + 7 ≤ 19
-4x ≤ 12
x ≥ -3 (reversed!)
```

**Example 6: Complex with Reversal**
```
-2(x - 3) > 10
-2x + 6 > 10
-2x > 4
x < -2 (reversed!)
```

### Compound Inequalities

**Example 7: "And" Compound**
```
-3 < 2x + 1 < 7
-4 < 2x < 6
-2 < x < 3
```

**Example 8: "Or" Compound**
```
x + 2 < -1 OR x - 3 > 4
x < -3 OR x > 7
```

**Example 9: Three-Part Inequality**
```
5 ≤ 3x - 4 ≤ 14
9 ≤ 3x ≤ 18
3 ≤ x ≤ 6
```

### Number of Solutions

**Example 10: One Solution (Equation)**
```
3x + 2 = 8
3x = 6
x = 2
```

**Example 11: No Solutions**
```
2x + 3 = 2x + 5
3 = 5 (false)
No solutions
```

**Example 12: Infinitely Many Solutions**
```
x + 3 = x + 3
3 = 3 (always true)
All real numbers
```

**Example 13: Inequality with No Solutions**
```
2x + 3 < 2x + 1
3 < 1 (false)
No solutions
```

**Example 14: Inequality - All Solutions**
```
x + 5 > x + 2
5 > 2 (always true)
All real numbers are solutions
```

### Application Examples

**Example 15: Budget Constraint**
```
Cost per item: $15
Budget: $200
15x ≤ 200
x ≤ 13.33
Maximum 13 items
```

**Example 16: Minimum Score**
```
Test average needed: ≥ 80
Scores: 75, 82, 78, x
(75 + 82 + 78 + x)/4 ≥ 80
235 + x ≥ 320
x ≥ 85
```

**Example 17: Temperature Range**
```
Celsius to Fahrenheit: F = 9/5C + 32
Safe range: 68 ≤ F ≤ 77
68 ≤ 9/5C + 32 ≤ 77
36 ≤ 9/5C ≤ 45
20 ≤ C ≤ 25
```

**Example 18: Profit Inequality**
```
Revenue: 25x
Cost: 500 + 10x
Profit > 0: 25x - (500 + 10x) > 0
15x > 500
x > 33.33
Need to sell at least 34 units
```

**Example 19: Time Constraint**
```
Activity 1: 2x hours
Activity 2: 3y hours
Total time available: ≤ 20 hours
2x + 3y ≤ 20
If x = 5: 10 + 3y ≤ 20
y ≤ 10/3 (maximum 3.33 hours)
```

**Example 20: Distance Problem**
```
Speed: ≥ 50 mph
Time: 3 hours
Distance: d = rt
d ≥ 50(3)
d ≥ 150 miles

Key Tips for Success

For Equations: Always check your solution by substituting back
For Inequalities: Remember to reverse the sign when multiplying/dividing by negatives
For Systems: Choose the method that makes coefficients easiest to eliminate
For Graphs: Practice identifying slopes and intercepts quickly
For Word Problems: Define variables clearly before setting up equations

With these comprehensive examples and strategies, you're well-equipped to tackle any linear algebra problem on the SAT!