**Middle School Number Sense Lesson 80: Solving for X with Two Variables**

Today's lesson (which can also be called

**solving a system of equations**) finishes out our "once through" of the 10 most common concepts for every 10 questions on middle school tests.

*In case you're wondering, on this year's tests, there have been*

**232 different concepts**tested in middle school. I have covered**140 of them (60%)**on this blog. But I'm covering the most frequently appearing questions. If you count the total # of questions (out of 1280) I've covered, it's**1035, or over 80%**. What can I say, except**You're Welcome**?Today's concept appeared

**8 times**last year, with a median placement at

**question # 79**.

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**Number Dojo Level: 271**

There are several ways to solve two equations with two unknowns. (See a thorough discussion by Khan Academy

**here**). I prefer to

**add the two equations together**. This works best when one variable has a positive value in one equation and a negative value in another. If the signs are the same on each pair of the same variable, we can

**subtract**one equation from the other, but this requires extra careful attention.

**How to Solve:**

- Ensure that the variables are in the same order (on the same side of the equation) and the constants are on the same side as each other. (If not, arrange them mentally so they are).
- Note which variable is being asked for, and try to eliminate the other. (If the variable being asked for is the easier one to eliminate, go ahead and solve for the other and then plug it back in to solve for the first).
- Add the equations together, or multiply one equation by some integer (positive or negative) and them add them together.
- Solve for the remaining variable. If necessary, use this solution to solve for the other variable.

We'll look at several examples to explain these better.

**Example 1: If 3x - y = 5 and 4x + y = 9, then x = ___**

- Ensure that the variables and constants are aligned--which they are.
- Notice that the variable being asked for is
**x**, and the easier variable to eliminate is**y**. (y has the same coefficient in both equations, but with opposite signs). This is ideal. - Add the equations together. We get
**3x + 4x - y + y = 5 + 9**, which simplifies to**7x = 14**. - Divide both sides by 7 and get
**x = 2**.

**Example 2: If 5x - y = 19 and x + y = 11, then x = ___**

- Ensure that the variables and constants are aligned--which they are.
- Notice that the variable being asked for is
**x**, and the easier variable to eliminate is**y**. (y has the same coefficient in both equations, but with opposite signs). This is ideal. - Add the equations together. We get
**5x + x - y + y = 19 + 11**, which simplifies to**6x = 30**. - Divide both sides by 6 to get
**x = 5**.

**Example 3: If 2x - 3y = 4 and x + y = 7, then x = ___**

- Ensure that the variables and constants are aligned--which they are.
- Notice the variable we need is
**x**, so we need to eliminate**y**. But we need to multiply the 2nd equation by**3**to eliminate the y. It will be 3(x + y = 7), or**3x + 3y = 21**. - Now add the first equation to the modified 2nd equation.
**2x + 3x - 3y + 3y = 4 + 21**, which simplifies to**5x = 25**. - Divide both sides by 5 to get
**x = 5**.

**Example 4: If 4x - 3y = 17 and 2x + 3y = 13, then y = ___**

- Ensure the variables and constants are aligned, which they are.
- Notice the variable we need is
**y**, which is also the easiest to eliminate. Go ahead and solve for x first by adding the equations together, and then plug the found x in to get y. - We get
**4x + 2x - 3y + 3y = 17 + 13**, which simplifies to**6x = 30**. This makes**x = 5**, which we can now plug in to one of the equations. (I think the 2nd one is easier). - 2(5) + 3y = 13, so 10 + 3y = 13, so 3y = 3, so
**y = 1**.

**Example 5: If 3x + 2y = 11 and x + y = 4.5, then x = ___**

- Ensure the variables and constants are aligned, which they are.
- Notice the variable we need is
**x**, so we need to eliminate**y**. But we need to multiply the 2nd equation by**-2**to eliminate the y. It will become -2(x + y = 4.5), or**-2x - 2y = -9**. - Now add the first equation to the modified 2nd equation.
**3x - 2x + 2y - 2y = 11 - 9**, which simplifies to**x = 2**.

**Here's a free worksheet to help you practice SolveX2var:**

solvex2var.pdf |

**Up Next for Middle School: MultDistribPr**