Chapter 4

Working with Two Equations Together

Section 4.1: Understanding Solutions of Simultaneous Linear Equations

In this chapter, we solve pairs of equations that go together. We call these simultaneous linear equations. "Simultaneous" is a big word. It just means "at the same time."

A linear equation looks like this: Ax + By = C. Here A, B, and C are just numbers. They can be positive or negative.

When we have two of these equations at once, we ask: what numbers can we put in for x and y to make both equations true at the same time? Those numbers are called the solution.

For example, look at these two equations:

• x + 2y = 10
• x − 3y = 0

If x = 6 and y = 2, both equations are true! So x = 6, y = 2 is the solution.

In this chapter, we will learn ways to find these solutions. We will use algebra (working with numbers and letters) and graphs (drawing pictures of the equations).

Example 1: The Ages of My Siblings

Let's say x = 21 and y = 17. These are the ages of my two older brothers or sisters. That's easy — there's only one answer!

But what if I want to make it trickier? I could just tell you their ages add up to 38. That's not enough — many pairs of numbers add up to 38!

So I give you one more clue: the difference between their ages is 4.

Now we can write this as two equations:

• x + y = 38 (the sum of their ages)
• x − y = 4 (the difference of their ages)

Can we find the ages now? Yes! We just need to figure out how to work backward from the sum and difference to find the original numbers.

Example 2: A Number Game

Here's a fun game:

1. Pick two numbers.
2. Double one number and add the other. Tell me the answer.
3. Now switch the numbers and do the same thing. Tell me that answer too.

Someone tells me: 30 and 27. I quickly say, "Your numbers were 8 and 11!"

How did I do that so fast? Let's figure it out.

Solving Example 1 (the ages):

We have:

• x + y = 38
• x − y = 4

If we add these two equations together:
(x + y) + (x − y) = 38 + 4

This simplifies to 2x = 42. So x = 21!

Now let's subtract the equations:
(x + y) − (x − y) = 38 − 4

This simplifies to 2y = 34. So y = 17!

Solving Example 2 (the number game):

The two numbers are x and y.

• Doubling the first and adding the second: 2x + y = 30
• Doubling the second and adding the first: 2y + x = 27

Let's add these:
(2x + y) + (2y + x) = 57

This simplifies to 3(x + y) = 57, so x + y = 19.

Now let's subtract:
(2x + y) − (2y + x) = 30 − 27

This simplifies to x − y = 3.

Now we know the sum (19) and the difference (3) — just like Example 1! Using the same steps, we find x = 11 and y = 8.

The Big Idea

These examples show something important. We can change one pair of equations into another pair. The answer (solution) stays the same! We do this to either hide the numbers (make a puzzle) or find the numbers (solve the puzzle).

Here are the tools (moves) we can use:

a. Add equal things to equal things.
(x = 21, y = 17 became x + y = 38)

b. Subtract equal things from equal things.
(x = 21, y = 17 became x − y = 4)

c. Multiply both sides by the same number (not zero).
(For 2y = 34, we multiplied both sides by 1/2 to get y = 17.)

Using these moves, we can change our equations without changing the answer. Remember: since we have two unknown numbers, we always need two equations. If I only tell you the sum of two numbers, that's not enough. I need to give you one more clue too — like the difference.

One More Tool: Substitution

d. Replace something in one equation with an equal thing from the other equation.

Example 3: Counting Marbles

Lovasz has 5 marbles more than double what Tonio has. Together, they have 107 marbles. How many marbles does Lovasz have?

Solution: Let's use letters. L = Lovasz's marbles. T = Tonio's marbles.

"Lovasz has 5 more than double Tonio's" means: L = 5 + 2T

"Together they have 107" means: L + T = 107

Since L equals 5 + 2T, we can swap it into the second equation:

5 + 2T + T = 107

Now we solve like we learned in Chapter 1. Combine the T's:

3T + 5 = 107

Subtract 5 from both sides:

3T = 102

So T = 34. Tonio has 34 marbles!

Now let's find L. Put T = 34 back into the first equation:

L = 5 + 2(34) = 5 + 68 = 73

Lovasz has 73 marbles!

Check: 73 + 34 = 107. ✓ It works!

Solving with Graphs

Now let's look at these equations using pictures called graphs.

Example 4: Finding Where Two Lines Cross

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

Solution: The first line has a slope of −3 (slope tells us how steep a line is). The second line has a slope of −1/3. Since the slopes are different, the lines are not parallel (they aren't running side-by-side forever). That means they must cross somewhere!

We can graph both lines using their starting points on the y-axis: (0, 7) for the first line, and (0, 5/3) for the second.

When we draw both lines, they cross at the point (2, 1).

Let's check: Does x = 2, y = 1 work in both equations?

3(2) + 1 = 7 ✓
2 + 3(1) = 5 ✓

Yes! When two lines cross, that crossing point is the solution to both equations at once. That's what it means to "solve" simultaneous equations!

Example 5: Another Crossing Point

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

Solution: Again, the slopes are different (1/2 and −2/5). So the lines are not parallel, and they must cross.

When we draw both lines, they cross at (12, 2).

Since this point is on both lines, x = 12 and y = 2 makes both equations true.

Since there's only one crossing point for two non-parallel lines, this is the only solution.

What If the Lines Never Cross?

Example 6: Parallel Lines

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

Solution: These two lines have different starting points on the y-axis: (0, 2) and (0, 4). But they have the same slope: −2/5.

Since they have the same slope but different starting points, the lines are parallel. Parallel lines never cross!

This means there is no solution — no values of x and y work for both equations. The graph shows this clearly: the lines run side by side and never touch.

Example 7: When Lines Are Exactly the Same

Now look at: 2x + 5y = 20 and 4x + 10y = 40.

These two equations actually describe the same line! They have the same slope and the same starting point.

If we rewrite both equations in slope-intercept form, they both become: y = −(2/5)x + 4

Since it's really just one line, every point on that line is a solution! There are infinitely many solutions.

Notice something: one equation is just a multiple of the other (divide the second equation by 2, and you get the first one). This always happens when a pair of equations has many solutions.

Summary: Three Possible Outcomes

When you have two linear equations, there are three things that can happen:

1. Different slopes. The lines cross at exactly one point. That point is the one and only solution.

2. Same slope, different lines. The lines are parallel and never cross. There is no solution.

3. Same slope, same line. The two equations describe the exact same line. Every point on the line is a solution — there are endless solutions!

So remember: if two lines have different slopes, they cross at one point. If two lines have the same slope, they are parallel — meaning either they never cross (no solution) or they are actually the same line (many solutions).

Looking Closer at the Graphs

Let's go back to Example 5: x − 2y = 8 and 2x + 5y = 34.

What happens if we add these two equations together? We get:

3x + 3y = 42

And if we subtract the first equation from the second, we get:

x + 7y = 26

If we draw these two new equations on the same graph as before, something interesting happens...