Math Problem Statement

Write a two column proof for the given equation

Solve the equations: (1) m / -3 + 10 = -1 and (2) 3x - 2 = 2(x + 5)

Solution

Let's analyze the two equations given in the image and write a two-column proof for each.

Problem 3:

m−3+10=−1\frac{m}{-3} + 10 = -1 We aim to solve for mm. Here is the two-column proof:

StepReason
1. m−3+10=−1\frac{m}{-3} + 10 = -1Given equation
2. m−3=−1−10\frac{m}{-3} = -1 - 10Subtract 10 from both sides
3. m−3=−11\frac{m}{-3} = -11Simplification
4. m=−11×−3m = -11 \times -3Multiply both sides by -3
5. m=33m = 33Simplification

Problem 4:

3x−2=2(x+5)3x - 2 = 2(x + 5) We aim to solve for xx. Here is the two-column proof:

StepReason
1. 3x−2=2(x+5)3x - 2 = 2(x + 5)Given equation
2. 3x−2=2x+103x - 2 = 2x + 10Distribute 2 on the right-hand side
3. 3x−2x=10+23x - 2x = 10 + 2Subtract 2x2x from both sides
4. x=12x = 12Simplification

Let me know if you have any questions or need further details!

Here are 5 related questions to expand understanding:

  1. How do you solve an equation with fractions like in Problem 3?
  2. What is the distributive property and how is it used in Problem 4?
  3. Can you solve the equation n5+6=2\frac{n}{5} + 6 = 2?
  4. What happens if there are variables on both sides of the equation?
  5. How do you solve multi-step equations in general?

Tip: Always isolate the variable by reversing operations in the opposite order of operations (PEMDAS).

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Math Problem Analysis

Mathematical Concepts

Algebra
Solving Linear Equations

Formulas

ax + b = c
Distributive Property

Theorems

Properties of Equality
Distributive Property

Suitable Grade Level

Grades 6-8

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