**Middle School Number Sense Lesson 37: Solving for X**

One of the most frequent concepts tested in number sense is

**solving basic equations**. On middle school tests I counted 59 of these questions last year, which I divided into 9 distinct concepts. This most basic version (without exponents, word problems, square roots, 2nd variables, etc.) was tested

**19**

**times**, with the median placement at

**question # 39**.

**Number Dojo Level: 131**

For a thorough study of solving for unknown variables, please see Khan Academy's

**website**. I will give a simple overview of this topic here. Here are a few definitions you need to understand before moving forward:

**Variable**: the unknown, represented by a letter of the alphabet. We are solving for this.**Constant**: the number seen by itself (on the same side of the equation as, but not attached to, the variable).**Coefficient**: the number attached to the variable.

**How to Solve:**

- If there is a
**constant**in the equation, add or subtract it from both sides of the equation to isolate the variable & coefficient. - If there is a
**coefficient**, divide both sides of the equation by the coefficient to isolate the variable.

**Example 1: If 4x - 7 = 25, then x = ___**

**7**is the constant. Add 7 to both sides of the equation: 4x - 7**+ 7**= 25**+ 7**. You are left with**4x = 32**.**4**is the coefficient. Divide both sides of the equation by 4. 4x**÷ 4**= 32**÷ 4**. You are left with**x = 8**. (You can always double-check yourself by plugging x back into the equation, but I don't recommend it on number sense tests--it's a waste of time. In this case, 4 x 8 - 7 = 25 --> 32 - 7 = 25 --> 25 = 25).

**Example 2: If 5 - 4x = 25, then x = ___**

**5**is the constant. Subtract 5 from both sides of the equation: 5 - 4x**- 5**= 25**- 5**. You are left with**-4x = 20**.**-4**is the coefficient. Divide both sides of the equation by -4. -4x**÷ -4**= 20**÷ -4**. You are left with**x = -5**.

**Example 3: If 9 - 7x = 11, then x = ___**

**9**is the constant. Subtract 9 from both sides of the equation: 9 - 7x**- 9**= 11**- 9**. You are left with**-7x = 2**.**-7**is the coefficient. Divide both sides of the equation by -7. -7x**÷ -7**= 2**÷ -7**. You are left with**x = -2/7**.

**Example 4: If 3x + 5 = 5x - 15, then x = ___**

*(Notice that we have the variable on each side of the equation. This will take an additional step)*. Let's start with the coefficient on the right side:**-15**. Add 15 to both sides of the equation. 3x + 5**+ 15**= 5x - 15**+ 15**. You are left with 3x + 20 = 5x.- Subtract 3x from both sides to isolate x on one side of the equation. 3x + 20
**- 3x**= 5x**- 3x**. You are left with**20 = 2x**. **2**is the coefficient. Divide both sides of the equation by 2. 20**÷ 2**= 2x**÷ 2**. You are left with**10 = x**.

**Example 5: If 4x + 9 = 23, then 4x - 1 = ___**

*(Notice that we aren't solving for x; we are solving for*.**4x - 1**. This is more of a logic problem than an algebraic equation)- Since the coefficient is the same in both equations, simply look at the difference in the constants (
**9 & -1)**. They are**10 apart**, so the answer must be**10 less than**the given equation.**23 - 10 = 13**, which is your answer. - To illustrate this better, we could have solved for
**4x**in the first equation. 4x + 9**- 9**= 23**- 9**. You are left with**4x = 14**. If 4x = 14, then 4x - 1 = 14 - 1 =**13**.

**Example 6: If 2x + 9 = 13, then 20x + 90 = ___**

*(Notice that we aren't solving for x; we are solving for***20x + 90**).- Also notice that the left side of the 2nd equation is exactly
**10 times**that of the 1st equation. Multiply 13 by 10 to get**130**, which is your answer. - To illustrate this better, we could have multiplied the entire first equation by 10: 10(2x + 9) = 10(13) --> 20x + 90 =
**130**, which is your answer.

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

solvex.pdf |

**Up Next for Middle School: SolveXExpo**