Systems of Linear Equations – SAT Math

📖 What Is a System of Linear Equations?

A system of linear equations is a set of two or more linear equations with the same variables. The solution is the point (x, y) that satisfies both equations.

2x + y = 10
x − y = 2

The solution to this system is the ordered pair (x, y) that makes both equations true.

📌 Key Concept

A solution to a system is a point where the two lines intersect. There are three possible outcomes:
1. One solution (lines intersect at one point)
2. No solution (lines are parallel, never intersect)
3. Infinite solutions (lines are identical, intersect everywhere)

⚡ Two Methods to Solve Systems

🔹 Method 1: Substitution

Substitution Method

Step 1: Solve one equation for one variable (isolate x or y).
Step 2: Substitute this expression into the other equation.
Step 3: Solve for the remaining variable.
Step 4: Substitute back to find the other variable.

🔹 Method 2: Elimination

Elimination Method

Step 1: Multiply equations if needed to get opposite coefficients.
Step 2: Add or subtract equations to eliminate one variable.
Step 3: Solve for the remaining variable.
Step 4: Substitute back to find the other variable.

📝 Solved Examples

Study these examples carefully. Each shows the step-by-step solution process.

EXAMPLE 1 — Substitution Method

Solve the system:
2x + y = 10
x − y = 2

Step 1: Solve the second equation for x: x = y + 2

Step 2: Substitute into the first equation: 2(y + 2) + y = 10

Step 3: Simplify: 2y + 4 + y = 10 → 3y = 6 → y = 2

Step 4: Substitute back: x = 2 + 2 = 4

✅ Solution: x = 4, y = 2

💡 Tip: Check: 2(4) + 2 = 10 ✓ and 4 − 2 = 2 ✓

EXAMPLE 2 — Elimination Method

Solve the system:
3x + 2y = 12
x − 2y = 8

Step 1: Notice the y terms are +2y and −2y (opposites!)

Step 2: Add the equations: (3x + 2y) + (x − 2y) = 12 + 8

Step 3: Simplify: 4x = 20 → x = 5

Step 4: Substitute back: 5 − 2y = 8 → −2y = 3 → y = −1.5

✅ Solution: x = 5, y = −1.5

💡 Tip: Check: 3(5) + 2(−1.5) = 15 − 3 = 12 ✓ and 5 − 2(−1.5) = 5 + 3 = 8 ✓

EXAMPLE 3 — Elimination (Multiply First)

Solve the system:
2x + 3y = 13
4x + y = 11

Step 1: Multiply the second equation by −3 to eliminate y: −12x − 3y = −33

Step 2: Add to first equation: (2x + 3y) + (−12x − 3y) = 13 + (−33)

Step 3: Simplify: −10x = −20 → x = 2

Step 4: Substitute back: 4(2) + y = 11 → 8 + y = 11 → y = 3

✅ Solution: x = 2, y = 3

💡 Tip: Check: 2(2) + 3(3) = 4 + 9 = 13 ✓ and 4(2) + 3 = 11 ✓

EXAMPLE 4 — No Solution

Solve the system:
2x + 3y = 7
4x + 6y = 10

Step 1: Multiply first equation by 2: 4x + 6y = 14

Step 2: Compare: 4x + 6y = 14 vs 4x + 6y = 10

Step 3: These are parallel lines (same slope, different y-intercept)

Step 4: No solution (lines never intersect)

✅ No solution

💡 Tip: When equations are multiples of each other but constants are different → no solution.

EXAMPLE 5 — Infinite Solutions

Solve the system:
2x + 3y = 7
4x + 6y = 14

Step 1: Multiply first equation by 2: 4x + 6y = 14

Step 2: This is identical to the second equation

Step 3: The two lines are the same line

Step 4: Infinite solutions (every point on the line is a solution)

✅ Infinite solutions

💡 Tip: When one equation is a multiple of the other with the same constant → infinite solutions.

🧪 Practice Questions

Solve each system. Click "Show Answer" to see the full solution.

Question 1

Solve the system:
x + y = 7
x − y = 1

A) x = 3, y = 4

B) x = 4, y = 3

C) x = 5, y = 2

D) x = 2, y = 5

✓ Answer: B

Add equations: 2x = 8 → x = 4 → y = 3

📝 Solution: x + y = 7, x − y = 1 → Add: 2x = 8 → x = 4 → y = 3

Question 2

Solve the system:
2x + y = 8
x − y = 1

A) x = 2, y = 4

B) x = 4, y = 2

C) x = 3, y = 2

D) x = 2, y = 3

✓ Answer: C

From x − y = 1 → x = y + 1. Substitute: 2(y + 1) + y = 8 → 3y = 6 → y = 2 → x = 3

📝 Solution: x = y + 1 → 2(y + 1) + y = 8 → 3y = 6 → y = 2 → x = 3

Question 3

Solve the system:
3x + 2y = 11
x + y = 4

A) x = 1, y = 3

B) x = 2, y = 2

C) x = 4, y = 0

D) x = 3, y = 1

✓ Answer: D

From x + y = 4 → y = 4 − x. Substitute: 3x + 2(4 − x) = 11 → 3x + 8 − 2x = 11 → x = 3 → y = 1

📝 Solution: y = 4 − x → 3x + 8 − 2x = 11 → x = 3 → y = 1

Question 4

Solve the system:
2x − 3y = 4
x + 2y = 9

A) x = 5, y = 2

B) x = 3, y = 3

C) x = 4, y = 1

D) x = 6, y = 0

✓ Answer: A

From x + 2y = 9 → x = 9 − 2y. Substitute: 2(9 − 2y) − 3y = 4 → 18 − 7y = 4 → y = 2 → x = 5

📝 Solution: x = 9 − 2y → 18 − 4y − 3y = 4 → −7y = −14 → y = 2 → x = 5

Question 5

Solve the system:
4x + y = 10
2x − y = 2

A) x = 1, y = 6

B) x = 3, y = 2

C) x = 4, y = −6

D) x = 2, y = 2

✓ Answer: D

Add equations: 6x = 12 → x = 2 → y = 2

📝 Solution: (4x + y) + (2x − y) = 10 + 2 → 6x = 12 → x = 2 → y = 2

Question 6

Solve the system:
3x + 2y = 16
5x − 2y = 8

A) x = 2, y = 5

B) x = 3, y = 3.5

C) x = 4, y = 2

D) x = 5, y = 0.5

✓ Answer: B

Add equations: 8x = 24 → x = 3 → 3(3) + 2y = 16 → 2y = 7 → y = 3.5

📝 Solution: (3x + 2y) + (5x − 2y) = 16 + 8 → 8x = 24 → x = 3 → y = 3.5

Question 7

Solve the system:
2x + 5y = 19
3x − 2y = 0

A) x = 1, y = 3.4

B) x = 3, y = 2.6

C) x = 2, y = 3

D) x = 4, y = 2.2

✓ Answer: C

From 3x − 2y = 0 → 3x = 2y → y = 1.5x. Substitute: 2x + 5(1.5x) = 19 → 2x + 7.5x = 19 → 9.5x = 19 → x = 2 → y = 3

📝 Solution: y = 1.5x → 2x + 7.5x = 19 → 9.5x = 19 → x = 2 → y = 3

Question 8

How many solutions does this system have?
2x + 4y = 8
x + 2y = 4

A) One solution

B) No solution

C) Two solutions

D) Infinite solutions

✓ Answer: D

2x + 4y = 8 is 2(x + 2y = 4). Both equations are identical → infinite solutions.

📝 Solution: 2x + 4y = 8 = 2(x + 2y) → Same line → Infinite solutions

Question 9

How many solutions does this system have?
3x + 2y = 6
3x + 2y = 10

A) One solution

B) No solution

C) Infinite solutions

D) Two solutions

✓ Answer: B

Same left side (3x + 2y), different right side (6 vs 10) → parallel lines → no solution.

📝 Solution: Same slope, different y-intercept → Parallel → No solution

Question 10

Solve the system:
5x + 3y = 29
2x − y = 5

A) x = 4, y = 3

B) x = 3, y = 4

C) x = 5, y = 2

D) x = 2, y = 5

✓ Answer: A

From 2x − y = 5 → y = 2x − 5. Substitute: 5x + 3(2x − 5) = 29 → 5x + 6x − 15 = 29 → 11x = 44 → x = 4 → y = 3

📝 Solution: y = 2x − 5 → 5x + 6x − 15 = 29 → 11x = 44 → x = 4 → y = 3

🎉 WELL DONE!

You've completed the Systems of Linear Equations lesson. You now know how to solve two-variable systems using substitution and elimination methods.

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