Systems of Linear Equation

1. Understanding Systems of Linear Equations

A system of linear equations is when we have two or more equations with the same variables, and we need to find values that satisfy ALL equations at the same time.

A solution to a system is a point (or pair of values) where both equations are true

What is a system?

System of equations

y = 2x + 1

y = -x + 4

We need to find the values of x and y that make BOTH equations true at the same time.

The solution is (x,y) = (1,3) because:

First equation: 3 = 2(1) + 1 ✓ True
Second equation: 3 = -(1) + 4 ✓ True

1.1 Types of Solutions

A system of equations can have:

One solution: The lines intersect at exactly one point
No solution: The lines are parallel (never intersect)
Infinite solutions: The lines are the same (identical equations)

2. Method 1: Substitution Method

The substitution method works by solving one equation for one variable, then substituting that expression into the other equation.

2.1 Steps for Substitution

Choose an equation and solve for one variable
Substitute that expression into the other equation
Solve for the remaining variabke
Substitute back to find the other varibale
Write the solution as an ordered pair (x,y)
Check your solution in both original equations

Substitution example 1

Solve the system:
y = x + 2

2x + y = 8

Step 1: The first equation is already solved for y: y = x + 2 Step 2: Substitute into the second equation:

2x + ( x + 2 ) = 8

Step 3: Solve for x:

2x + x + 2 = 8

3x + 2 = 8

3x = 6

x = 2

Step 4: Find y by substituting x = 2 into the first equation:

y = 2 + 2 = 4

Step 5: Solution (2,4)

Step 6: Check:

Equation 1: 4 = 2+2
Equation 2: 2(2) + 4 = 4 + 4 = 8

Substitution Example 2

Solve the system:

x + y = 5

2x - y = 4

Step 1: Solve the first equation for y:

x + y = 5

y = 5 - x

Step 2: Substitute into the second question:

2x - (5 - x) = 4

Step 3: Solve for x:

2x - 5 + x = 4

3x - 5 = 4

3x = 9

x = 3

Step 4: Find y:

y = 5-3 = 2

Step 5: Solution: (3, 2)

Step 6: Check:

Equation 1: 3 + 2 = 5
Equation 2: 2(3) - 2 = 6 - 2 = 4

Practice Problem

Solve each system using substitution:

y = 2x and x + y = 9
x = y - 3 and 2x + y = 12
y = -x + 5 and 3x + y = 11

3. Method 2: Elimination Method

The elimination method works by adding or subtracting equations to eliminate one variable making it easier to solve.

3.1 Steps for Elimination

Arrange both equations in standard form (ax + by = c)
Look for a way to make one variable's coefficients opposites
Multiply one or both equations by a number if needed
Add the equations together to eliminate one variable
Solve for the remaining variable
Substitute back to find the other variable
Check your solution in both original equations

Elimination Example 1 - Variables Already Opposites

Solve the system:

2x + y = 7

2x - y = 5

Step 1: Already in standard form

Step 2-3: Notice that y and -y are opposites! No multiplication needed.

Step 4: Add the equations:

(2x + y) + (2x - y) = 7 + 5

4x = 12

Step 5: Solve for x:

x = 3

Step 6: Substitute into the first equation

2(3) + y = 7

6 + y = 7

y = 1

Solution: (3,1)

Step 7: Check:

Equation 1: 2(3) + 1 = 6 + 1 = 7
Equation 2: 2(3) - 1 = 6 - 1 = 5

Elimination Example 2 - Need to Multiply

Solve the system:

3x + 2y = 11

x + 2 y = 7

Step 1: Already in Standard Form

Step 2: To eliminate x, multiply the second term to make x coefficients opposites:

Equation 1: 3x + 2y = 11

Equation 2 (multiplied by -3): -3x - 6x = -21

Step 3: Multiplying Done

Step 4: Add the equations:

(3x + 2y) + (-3x - 6y) = 11 + (-21)

-4y = 10

Step 5 solve for y:

y = ^-10⁄_-4 = ¹⁰⁄₄ = ⁵⁄₂ = 2.5

Step 6: Substitute into equation 2:

x + 2(2.5) = 7

x + 5 = 7

x = 2

Solution (2, 2.5)

Step 7: Check:

Equation 1: 3(2) + 2(2.5) = 6 + 5 = 11
Equation 2: 2 + 2(2.5) = 2 + 5 = 7

Elimination Example 3 - Multiply Both Equations

2x + 3y = 8

3x + 2y = 7

Step 1: Already in standard form

Step 2-3: To eliminate x:

Multiply first equation by 3: 6x + 9y = 24
Multiply second equation by -2: 6x - 4y = -14

Step 4: Add equations:

(6x + 9y) + (-6x - 4y) = 24 + (-14)

5y = 10 Step 5: Solve for y:

y = 2

Step 6" Substitute into the first equation:

2x + 3(2) = 8

2x + 6 = 8

2x = 2

x = 1

Solution: (1,2)

Step 7: Check:

Equation 1: 2(1) + 3(2) = 2 + 6 = 8
Equation 2: 3(1) + 2(2) = 3 + 4 = 7

Practice Problem

Solve each system using elimination:

x + y = 10 and x - y = 4
2x + y = 7 and x - y = 2
2x + 3y = 13 and x + 2y = 8

4. Method 3: Graphing Method

The graphing method Involves plotting both equations on a coordinate plane. The point(s) where the lines intersect is the solution

4.1 Steps for Graphing

Rewrite both equations in slope intersept form: y = mx + b
Identify slope (m) and y-intercept (b) for each
Plot both lines on the same graph
Find the intersection point(s)
The intersection point (x,y) is your solution

Graphing Example

Solve the system by graphing:

y = 2x - 1

y = -x + 2

Step 1: Both Equations are already in slope-intercept form

Step 2:

First line: m = 2, b = -1 (slope is 2, y intercept is -1)
Second line: m = -1, b = 2 (slope is -1, y intercept is 2)

Step 3: Plot the lines

First line passes through (0, -1) and (1,1)
Second line passes through (0,2) and (1,1)

Step 4-5: The lines intersect at the point (1,1)

Solution: (1,1)

Verify:

Equation 1: 1 = 2(1) - 1 = 1
Equation 2: 1 = -(1) + 2 = 1

4.2 Converting to Slope-Intercept Form

If your equations aren't in y = mx + b form, solve for y first.

Converting to Slope-Intercept Form

Converting 2x + y = 5 to slope-intercept form:

2x + y = 5

y = -2x + 5

Now we can see m = -2 and b = 5

Practice Problem

Solve each system by graphing (describe the intersection):

y = x and y = -x + 2
y = 2x + 1 and y = x - 1

5. Word Problems Using Systems

Many real-world situations involve systems of equations. The key is translating English into mathematical equations.

5.1 Steps to Solve Word Problems

Define your variables (say what x and y represent)
Write two equations based on the problem
Solve the system using substitution or elimination
Check that your answer makes sense in the context
Write your answer in a complete sentence

Word Problem 1: Purchase Problem

Maria bought 3 notebooks and 2 pens for $13. James bought 2 notebooks and 4 pens for $16. Find the price of each item.

Step 1: Define variables

Let x = price of one notebook
Let y = price of one pen

Step 2: Write equations

Maria's purchase: 3x + 2y = 13
James's purchase: 2x + 4y = 16

Step 3: Solve using elimination

Multiply first equation by -2 and second by 3:

-6x - 4y = -26
6x + 12y = 48

Add them: 8y = 22, so y = 2.75

Substitute: 3x + 2(2.75) = 13

3x + 5.5 = 13
3x = 7.5
x = 2.5

Step 4: Check: Does this make sense?
3 notebooks (3 × $2.50 = $7.50) plus 2 pens (2 × $2.75 = $5.50) = $13 ✓

Step 5: Answer: A notebook costs $2.50 and a pen costs $2.75.

Word Problem 2: Number Problem

The sum of two numbers is 24. The difference between them is 6. Find the numbers.

Step 1: Define variables

Let x = the larger number
Let y = the smaller number

Step 2: Write equations

Sum: x + y = 24
Difference: x - y = 6

Step 3: Solve using elimination

Add the equations:
(x + y) + (x - y) = 24 + 6
2x = 30
x = 15

Substitute: 15 + y = 24, so y = 9

Step 4: Check:

Sum: 15 + 9 = 24 ✓
Difference: 15 - 9 = 6 ✓

Step 5: Answer: The two numbers are 15 and 9.

Word Problem 3: Distance/Rate/Time

A fast train and a slow train leave the same station. The fast train travels at 80 mph and the slow train at 60 mph. How long before they are 150 miles apart?

Step 1: Define variables

Let t = time (in hours)
Let d_f = distance traveled by fast train
Let d_s = distance traveled by slow train

Step 2: Write equations

Fast train distance: d_f = 80t
Slow train distance: d_s = 60t
Difference in distance: d_f - d_s = 150

Step 3: Substitute and solve

80t - 60t = 150
20t = 150
t = 7.5

Step 4: Check: Fast train goes 80 × 7.5 = 600 miles. Slow train goes 60 × 7.5 = 450 miles. Difference = 150 ✓

Step 5: Answer: After 7.5 hours, the trains will be 150 miles apart.

Practice Problem

Solve these word problems:

The sum of two numbers is 20. One number is 4 more than the other. Find the numbers.
Tickets cost $5 for adults and $3 for children. If 40 tickets were sold for a total of $160, how many adult tickets and how many children's tickets were sold?
A rectangle has a perimeter of 36 inches. The length is 2 inches more than the width. Find the dimensions.

8. Answer Key

Substitution:

1. (3, 3) - Substitute y = 2x - 3 into x + y = 6 to get x + 2x - 3 = 6, so 3x = 9, x = 3, y = 3

2. (5, 3) - Substitute x = y + 2 into 2x - y = 7 to get 2(y + 2) - y = 7 ⇒ 2y + 4 - y = 7 ⇒ y = 3, then x = 5

Elimination:

3. (2, 1) - Add equations: 3x = 6, so x = 2. Substitute: 2 + y = 5, y = 1

4. (4, 0) - Add equations: 4x = 16, so x = 4. Substitute: 3(4) + 2y = 12, so y = 0

Graphing:

5. (1, 2) - Lines intersect where x + 1 = -x + 3, so 2x = 2, x = 1, y = 2

6. (2, 2) - Lines intersect where 2x - 2 = -x + 4, so 3x = 6, x = 2, y = 2

Word Problems:

7. Answer: The numbers are 19 and 11.

8. Answer: 4 apples and 4 oranges

9. Answer: Length is 15 inches, width is 10 inches

Systems of Linear Equations Reviewer

1. Understanding Systems of Linear Equations

1.1 Types of Solutions

2. Method 1: Substitution Method

2.1 Steps for Substitution

3. Method 2: Elimination Method

3.1 Steps for Elimination

4. Method 3: Graphing Method

4.1 Steps for Graphing

4.2 Converting to Slope-Intercept Form

5. Word Problems Using Systems

5.1 Steps to Solve Word Problems

6. Choosing the Best Method

7. Review Problems

Substitution Method:

Elimination Method:

Graphing Method:

Word Problems:

8. Answer Key

Substitution:

Elimination:

Graphing:

Word Problems: