Two-Column Proof for Solving Linear Equations

Math Problem Statement

Write a two-column proof for the following equations: 1) 8x - 10 = 3(6 - 2x) 2) 5n - 42 = 12n 3) m/-3 + 10 = -1 4) 3x - 2 = 2(x + 5)

Solution

Let's solve the given equations using a two-column proof format.

1. $8x - 10 = 3(6 - 2x)$

Steps	Reasons
$8x - 10 = 3(6 - 2x)$	Given
$8x - 10 = 18 - 6x$	Distribute the 3 across $(6 - 2x)$
$8x + 6x = 18 + 10$	Add $6x$ and $10$ to both sides
$14x = 28$	Simplify
$x = 2$	Divide both sides by 14

2. $5n - 42 = 12n$

Steps	Reasons
$5n - 42 = 12n$	Given
$5n - 12n = 42$	Subtract $12n$ from both sides
$-7n = 42$	Simplify
$n = -6$	Divide both sides by -7

3. $\frac{m}{-3} + 10 = -1$

Steps	Reasons
$\frac{m}{-3} + 10 = -1$	Given
$\frac{m}{-3} = -1 - 10$	Subtract 10 from both sides
$\frac{m}{-3} = -11$	Simplify
$m = 33$	Multiply both sides by -3

4. $3x - 2 = 2(x + 5)$

Steps	Reasons
$3x - 2 = 2(x + 5)$	Given
$3x - 2 = 2x + 10$	Distribute the 2 over $(x + 5)$
$3x - 2x = 10 + 2$	Subtract $2x$ and add 2 to both sides
$x = 12$	Simplify

Would you like more details on any step? Here are five related questions:

How do you verify a solution for an equation?
Can you explain how distributing works in algebra?
What is the reason for combining like terms in an equation?
How can you solve equations involving fractions?
How do you handle equations where variables appear on both sides?

Tip: Always perform the same operation on both sides of an equation to maintain balance!

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
Distributive Property
Combining Like Terms

Formulas

Distributive property: a(b + c) = ab + ac
Solving linear equations: ax + b = c

Theorems

Properties of equality: If a = b, then a + c = b + c and a - c = b - c
Distributive Property: a(b + c) = ab + ac

Suitable Grade Level

Grades 7-9

Related Recommendation

Algebra Proofs for Linear Equations: Solve and Verify

Two Column Proof for Solving 25 + 5(4 - x) = 5

Solving Linear Equations Step-by-Step

Solving Linear Equations and Identifying No-Solution Equations

Algebra Equation Solving Worksheet - Linear Equations