Chapter 4

Simultaneous Linear Equations

Section 4.1: Understanding Solutions of Simultaneous Linear Equations

Sometimes we have two equations. We use them at the same time. This is called simultaneous equations.

We look for numbers that work in both equations. Those numbers are called the solution.

Look at this example. x + 2y = 10. Also, x − 3y = 0. We want one x and one y that work for both. It turns out x = 6 and y = 2 works for both!

We will learn ways to find these answers. We will use pictures and math steps.

Example 1: My Siblings' Ages

I know my two siblings' ages. I want to keep them secret. But I give clues.

Clue one: Their ages add up to 38.
That is not enough. Many numbers add to 38.

Clue two: The difference between their ages is 4.

Now we have two clues. We can write them as math:
x + y = 38
x − y = 4

Can we find their ages now? Yes we can!

Example 2: A Number Game

Pick two secret numbers.

Double one number. Add the other number. Tell me the answer.

Now switch the numbers. Do it again. Tell me that answer.

I heard 30 and 27. I say, "Your numbers are 8 and 11!"

How did I know? We can add the two clues together. We can also subtract one clue from the other. This helps us find the secret numbers.

Finding the Trick

In both examples, we start with two numbers. Then we mix them up with math.

To unscramble them, we can:

• Add the two equations together.
• Subtract one equation from the other.
• Multiply both sides by the same number.

These steps help us find x and y. We always need two equations to find two unknown numbers.

Example 3: Lovasz and Tonio's Marbles

Lovasz has 5 more marbles than twice Tonio's marbles.

Together they have 107 marbles.

We write this as math:
L = 5 + 2T
L + T = 107

We swap L for "5 + 2T" in the second equation:
5 + 2T + T = 107

We solve step by step. We get T = 34. Tonio has 34 marbles!

Then we find L = 5 + 2(34) = 73. Lovasz has 73 marbles!

We check: 73 + 34 = 107. Yes, it works!

Graphs and Equations

We can also draw pictures of equations. Each equation makes a line. Sometimes two lines cross. That crossing point is the answer to both equations!

Example 4: Two Lines Cross

Look at these equations:
3x + y = 7
x + 3y = 5

We draw both lines on a graph. They cross at one point: (2, 1).

We check it works:
3(2) + 1 = 7 ✓
2 + 3(1) = 5 ✓

The crossing point is the solution!

Example 5: Another Crossing Point

Look at these equations:
x − 2y = 8
2x + 5y = 34

We draw both lines. They cross at (12, 2).

Since the lines have different tilts (called slopes), they can only cross once. So this is the only answer.

Example 6: Lines That Never Cross

Look at these equations:
2x + 5y = 10
4x + 10y = 40

We draw both lines. They have the same tilt. That means they are parallel. Parallel lines never cross!

So there is no solution. No numbers work for both equations.

Example 7: Lines That Are the Same

Look at these equations:
2x + 5y = 20
4x + 10y = 40

These make the exact same line! One equation is just a bigger copy of the other.

Since the lines are the same, every point on the line is a solution. There are many, many answers!

Summary

When we have two equations, three things can happen:

1. Different tilts. The lines cross at one point. There is one solution.

1. Same tilt, different lines. The lines are parallel. They never cross. There is no solution.

1. Same tilt, same line. The lines are exactly the same. Every point works. There are many solutions.