Systems of Linear Equations

What is a System of Equations?

A system of equations is two or more equations with the same variables.

Example:

y = 2x + 1
y = −x + 4

Solution: The point (x, y) that makes BOTH equations true!

Graphing Method

Idea: Graph both lines and find where they intersect

The intersection point is the solution!

Example 1: Solve by Graphing

System:

y = x + 1
y = −x + 5

Step 1: Graph first equation y = x + 1

• y-intercept: (0, 1)
• Slope: 1 (up 1, right 1)

Step 2: Graph second equation y = −x + 5

• y-intercept: (0, 5)
• Slope: −1 (down 1, right 1)

Step 3: Find intersection

• Lines cross at (2, 3)

Step 4: Check in both equations

• Equation 1: 3 = 2 + 1 ✓
• Equation 2: 3 = −2 + 5 ✓

Answer: (2, 3)

Example 2: Solve Graphically

System:

y = 2x
y = x + 2

Graph both lines:

• y = 2x: Through origin, slope 2
• y = x + 2: y-intercept (0,2), slope 1

Intersection: (2, 4)

Check:

• 4 = 2(2) ✓
• 4 = 2 + 2 ✓

Answer: (2, 4)

Substitution Method

When graphing is difficult: Use algebra!

Steps:

1. Solve one equation for one variable
2. Substitute into other equation
3. Solve for remaining variable
4. Substitute back to find other variable
5. Check solution

Example 1: Use Substitution

System:

y = 3x − 2
2x + y = 8

Step 1: First equation already solved for y!

• y = 3x − 2

Step 2: Substitute into second equation

2x + (3x − 2) = 8

Step 3: Solve for x

2x + 3x − 2 = 8
5x − 2 = 8
5x = 10
x = 2

Step 4: Find y using y = 3x − 2

y = 3(2) − 2
y = 6 − 2
y = 4

Step 5: Check

• Equation 1: 4 = 3(2) − 2 ✓
• Equation 2: 2(2) + 4 = 8 ✓

Answer: (2, 4)

Example 2: Solve One Equation First

System:

x + y = 7
y = 2x + 1

Step 1: Second equation already solved for y

Step 2: Substitute y = 2x + 1 into first equation

x + (2x + 1) = 7
3x + 1 = 7
3x = 6
x = 2

Step 3: Find y

y = 2(2) + 1 = 5

Answer: (2, 5)

Example 3: Must Solve for Variable First

System:

2x + y = 10
x − y = 2

Step 1: Solve second equation for x

x − y = 2
x = y + 2

Step 2: Substitute into first equation

2(y + 2) + y = 10
2y + 4 + y = 10
3y + 4 = 10
3y = 6
y = 2

Step 3: Find x

x = 2 + 2 = 4

Answer: (4, 2)

Types of Solutions

One solution: Lines intersect at one point

• Different slopes

No solution: Lines are parallel

• Same slope, different y-intercepts
• Never intersect

Infinite solutions: Lines are the same

• Same slope, same y-intercept
• Every point on the line is a solution

Example: No Solution

System:

y = 2x + 1
y = 2x + 5

Both have slope 2, different y-intercepts

Parallel lines → No solution

Example: Infinite Solutions

System:

y = 3x + 2
2y = 6x + 4

Second equation: 2y = 6x + 4 → y = 3x + 2

Same equation! → Infinite solutions

Writing Systems from Word Problems

Example: Two Numbers

Problem: "Two numbers sum to 12. Their difference is 4."

Translate:

x + y = 12
x − y = 4

Solve using substitution:

• From equation 2: x = y + 4
• Substitute: (y + 4) + y = 12
• 2y + 4 = 12
• y = 4, x = 8

Answer: Numbers are 8 and 4

Real-World Applications

Business: Finding break-even point

• Revenue equation
• Cost equation
• Where they're equal = break-even

Mixture problems: Combining solutions

• Different concentrations
• Find amounts needed

Rate problems: Two objects moving

• When do they meet?
• Where do they meet?

Geometry: Finding dimensions

• Perimeter and area equations
• Solve for length and width

Example: Ticket Sales

Problem: Adult tickets $8, child tickets $5. Sold 50 tickets for $337.

System:

a + c = 50  (total tickets)
8a + 5c = 337  (total money)

Solve:

• From equation 1: a = 50 − c
• Substitute: 8(50 − c) + 5c = 337
• 400 − 8c + 5c = 337
• 400 − 3c = 337
• c = 21, a = 29

Answer: 29 adult, 21 child tickets

Choosing a Method

Graphing:

• Good for visualizing
• Less accurate
• Quick estimate

Substitution:

• When one equation is solved for a variable
• More accurate
• Works for all systems

Other methods (not covered here):

• Elimination
• Matrices

Practice