Grade 7: Solving Two-Step Equations

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Grade 7: Solving Two-Step Equations

Recall that when we apply the same mathematical operation (addition, subtraction, multiplication, division) to both sides of an equation we preserve the equality of the equation.

An equation that includes a variable, such as x, is called an algebraic equation. To solve the equation means to figure out the value of the variable that makes the equation true. To solve a two-step equation, we must apply two different mathematical operations. The order in which we apply the operations to solve the equation is the opposite of the order that we use to evaluate an expression.

Definition: An expression is a meaningful combination of variables numbers and symbols. There is no = sign in an expression.

Example: 4 + 5 is an expression.

If you are asked to evaluate the expression 4 + 5 for = 3, then we substitute 3 for the x in the equation,

4 + 5 Using order of operations

we first multiply 4 x 3

Definition: An equation equates two expressions to declare a relationship. An equation must include an = sign.

Example: 4 + 5 = 17 is an equation.

If you are asked to solve this equation, you must apply the math operations to both sides of the equation in the reverse order of the order of operations used to evaluate expressions. In this example, you would first subtract 5 from both sides of the equation, and then divide both sides by 4 to solve for .

4 + 5 = 17 First subtract 5 from both sides

and then add 5

You are left with the equation

4 = 12

Then divide both sides by 4

4 = 12

to evaluate the expression for = 3

To get the solution

You can check your solution to an equation by substituting your solution back into the original equation. If the equation balances (both sides of the equation are equal) then the solution to the equation is correct. Note that the value on both sides of the equation will not be the same as the solution that you are testing. The important thing is that there is the same number on both sides of the equation after you substitute your solution.

Example:

Solve

the

equation

6

-

2

=

7

Check your solution

First add 2 to both sides

6

-

2=

7

You are left with the equation

= 9

6

Substitute

into the original equation

6

-

2

=

7

Then multiply both sides by 6

6

= 9

To get the solution

Since both sides of the equation are equal after you substitute the solution, the solution is correct.

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Practice Solving Two-Step Equations and Check by substituting your solution into the original equation.*

1. 4x + 55 = 143

Check solution:

2. 72 = 2 y -14

Check solution:

3. 14w - 23 = 201

Check solution:

4. 72 - 9z = 198 5. -19a -18 = 191 6. 60 = 2b - 26

Check solution: Check solution: Check solution:

* 1. x=22 2. y=43 3. w=16 4. z=-14 5. a=-11 6. b=43

OntarioMathTeacher.ca Solving Two-Step Equations: Practice Questions - continued*

7. 11-14c = 1019

Check solution:

8. 561= 11d + 55 9. - 219 = 16 + 47 p

Check solution: Check solution:

10. - 400.5 = -18 - 9q

Check solution:

11. 16r + 66 = 370

Check solution:

12. 39 -16s = 311

Check solution:

* 7. c=-72 8. d=46 9. p=-5 10. q=42.5 11. r=19 12. s=-17

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Equations with a minus sign in front of the variable

Method 1: Treating the variable with a minus sign like a number. We used this approach in the practice questions for Solving One-Step Equation by addition and subtraction.

When there is a minus sign in front of the variable, for example 15 - = 11, we can add to both sides of the equation

and then continue to solve:

15 -

= 11

The - + cancel out and we are left with 15 = 11 +

Subtract 11 from both sides to get by itself. The solutions is 4 = which is the same as = 4

Method 2: Isolating the variable with the minus sign, and then dividing by -1.

Equations with a negative in front of the variable can be solved as two-step equations. Recall that - = -1 . In the example 15 - = 11, we start by subtracting 15 from both sides, then dividing by -1.

15 -

= 11

Subtract 15 from both sides and you are left with - = -4

Remember - = -1 so we can divide both sides by -1 to get the answer, = 4

Practice using Method 2:

1. 46 - = 20

2. 23 - = -16

3. 48 = 24 -

4. -37 - = 13

5. 37 = 41 -

6. 35 = 27 -

7. 61 = 34 -

8. 15 = 39 -

1. = 26 2. = 39 3. = -24 4. = -50 5. = 4 6. = -8 7. = -27 8. = 24

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More practice with a minus sign in front of the variable. Solutions

1. 49 - = 17

2. 43 - = -22

3. 68 = 84 -

4. -34 - = 19

5. 31 = 46 -

6. 31 = 26 -

7. 11 = 37 -

8. 17 = 29 -

9. 24 - = 16

10. 11 = 18 -

11. 25 - = 37

12. 26 - = -14

1. = 32 2. = 65 3. = 16 4. = -53 5. = 15 6. = -5 7. = 26 8. = 12 9. = 8 10. = 7 11. = -12 12. = 40

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