# Common Core Standards: Math

### Expressions and Equations 8.EE.C.7

7. Solve linear equations in one variable.

One of the really nice things about math is that there's always one right answer. None of that wishy-washy, subjective, personalized, "What do you think?" stuff in math. Let English teachers deal with all that. In math, if an answer is right, it's right. That's it. The end. No questions asked.

Right?

Well… no, not always. Sometimes in math, there are no right answers and sometimes there are an infinite number of answers to a problem.

Make sure to clean your students' brains off the walls after their heads explode.

#### Drills

1. Solve the equation 4x + 10 = 6x.

x = 5

When we subtract 4x from both sides, we get 10 = 2x. Dividing both sides by 2 gives us x = 5. Simply plugging in x = 1 or x = 10 into the equation can prove that those values don't work. (After all, 50 ≠ 60 and 14 ≠ 6, right?) If there exists a real number that works for x, then (D) must be incorrect also.

2. Solve the equation 2x + 1 = 15.

x = 7

We first need to subtract 1 from both sides, leaving 2x = 14. Dividing both sides by 2 gives us an answer of x = 7. Answer choices (A) and (B) result from forgetting to divide by 2 or adding 1 instead of subtracting 1. If x = 7 is the only value that works, then (D) can't be true.

3. Solve the equation x + 5 = x + 8.

No solution

When we subtract x to get all the x's together, we get 5 = 8. There's no real value of x that will make that true, so the answer is (D). The other three answers will give incorrect equations (8 = 11, 10 = 13, and 13 = 16), so we can be sure they aren't right.

4. Solve the equation 3x + 10 = 4x + 10.

x = 0

When we subtract 3x from both sides, we get 10 = x + 10. After subtracting 10 from both sides, we get an answer of 0 = x. This does not mean there's no answer! It means that only x = 0 will make the equation a true statement.

5. Solve the equation 4x + 8 = 7x + 8 – 3x.

x can equal all real numbers

First, combine the x values on the right to get 4x + 8 = 4x + 8. It should be clear from here, but if it isn't, then continue by subtracting 4x from both sides to get 8 = 8. It doesn't matter what x equals because 8 will always equal 8.

6. Solve the equation 5x + 10 = 10x.

x = 2

Subtract 5x from both sides to get 10 = 5x. If we divide both sides by 5, we should get x = 2 as our answer. All the other answers result from misinterpreting the equation or incorrectly combining terms.

7. For which of the following equations can x equal all real numbers?

7(2x + 3) = 11x + 3(x + 7)

If we try to solve each equation for x, we end up with 0 = 0 for (A), x = 3 for (B), x = -1 for (C), and 9 = 2 for (D). While (B) and (C) give single numbers as the answer, (D) is impossible, which means that there is no solution, and (A) is always true for any value of x. The only equation in which x can be all real numbers is (A).

8. For which of the following equations can x = 0 only?

4x + 8 = 8 – 4x

There's really no shortcut here. We have to reduce every single equation to see if we can solve for x = 0. Only in (A) does this happen because subtracting 8 from both sides gives us 4x = -4x. Adding 4x gives 8x = 0, or x = 0. The other answer choices give 8 = 8 (so x can be all real numbers), or x = 2. And since only (A) is the answer, (D) clearly isn't.

9. Which of the following equations has no solution?

4(x + 2) + x = 5(x + 3)

Solving each equation one by one should give us the answer. Properly solving (A) should give us x = 1, (B) should be x = -15, while (C) gives 8 = 15 and (D) gives 9 = 9. Since 9 will always be equal to itself, any value of x will work for it. On the other hand, no value of x will make 8 = 15, so (C) has no solution.

10. Which of the following equations has no solution?

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