Math Problem Statement

Use this as a guide : [Solve 10x = 2x − 16. Check: 10x = 2x − 16 10x = 2x − 16 =?

−2x −2x Add −2x to both sides. 10(−2) 2(−2) − 16 =?

8x = −16 −20 −4 − 16

−16

8x

8=?

= Divide both sides by 8. ]

Write the first step you would take to solve each equation.

  1. 3x + 2 = 7x

  2. −4x − 6 = −10x

  3. 15x + 7 = −3x Solve each equation. Check your answers.

  4. 4x + 2 = 5(x + 10)

  5. −10 + y + 3 = 4y − 13

  6. 3(t + 7) + 2 = 6t − 2 + 2t

use this as a guide: ["Solve −3x + 9 = 4x + 9 − 7x. Check any value of x: −3x + 9 = −3x + 9 Combine like terms. Try x = 4.

−3x +3x Add 3x to each side. −3x + 9 = 4x + 9 − 7x =?

9 = 9 ✔ True statement. −3(4) + 9 4(4) + 9 − 7(4) =?

The solution is the set of all real numbers. −12 + 9 16 + 9 − 28 =?

olve 2x + 6 + 3x = 5x − 10. Check any value of x: 2x + 6 + 3x = 5x − 10 Try x = 1.

5x + 6 = 5x − 10 Combine like terms. 2x + 6 + 3x = 5x − 10 =?

−5x −5x Add −5x to each side. 2(1) + 6 + 3(1) 5(1) − 10 =?

6 = −10 🗶 False statement. 2 + 6 + 3 5 − 10 =?

There is no solution. 11 −5 🗶 "]

Solve each equation.

  1. x + 2 = x + 4

  2. −2x + 8 = 2x + 4

  3. 5 + 3g = 3g + 5

  4. 5x − 1 − 4x = x + 7

  5. 2(f + 3) + 4f = 6 + 6

  6. 3x + 7 − 2x = 4x + 10

Solution

Let's go through the first step for each of the equations, then solve and check where requested.

First Steps:

  1. 3x + 2 = 7x

    • First step: Subtract 3x3x from both sides to get the x terms on one side.

  2. −4x − 6 = −10x

    • First step: Add 10x10x to both sides to move the x terms to one side.

  3. 15x + 7 = −3x

    • First step: Add 3x3x to both sides to move the x terms to one side.

Now let's solve and check the remaining equations:

  1. 4x + 2 = 5(x + 10)

    • First step: Distribute 55 on the right-hand side to simplify. 4x+2=5x+504x + 2 = 5x + 50
    • Subtract 4x4x from both sides: 2=x+502 = x + 50
    • Subtract 5050 from both sides: x=−48x = -48
    • Check: 4(−48)+2=5(−48+10)4(-48) + 2 = 5(-48 + 10) −192+2=5(−38)-192 + 2 = 5(-38) −190=−190✔ True statement.-190 = -190 \quad \text{✔ True statement.}

  2. −10 + y + 3 = 4y − 13

    • First step: Combine like terms on the left-hand side: −7+y=4y−13-7 + y = 4y - 13
    • Subtract yy from both sides: −7=3y−13-7 = 3y - 13
    • Add 1313 to both sides: 6=3y6 = 3y
    • Divide by 33: [ y = 2

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
Variable Solving
Equation Balancing

Formulas

ax + b = cx + d
Distributive Property a(b + c) = ab + ac
Combining Like Terms

Theorems

Addition Property of Equality
Subtraction Property of Equality

Suitable Grade Level

Grades 7-9

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