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Chapter 10: Problem 79

Solve each equation. See Sections 2.3 and 6.6 $$2 x-7=3(x-4)$$

Short Answer

Expert verified
The solution to the equation is \(x = 5\).

Step by step solution

01

Distribute the Right Side

We start by applying the distributive property to the expression on the right side of the equation. The equation is \( 2x - 7 = 3(x - 4) \). Distributing the \(3\) gives \( 2x - 7 = 3x - 12 \).
02

Move Variables to One Side

To get all variable terms on one side, subtract \(2x\) from both sides of the equation: \( 2x - 7 - 2x = 3x - 12 - 2x \). This simplifies to \( -7 = x - 12 \).
03

Isolate the Variable

Add \(12\) to both sides to isolate \(x\): \( -7 + 12 = x - 12 + 12 \). This simplifies to \( 5 = x \).
04

Verify the Solution

Substitute \(x = 5\) back into the original equation to check the solution: \( 2(5) - 7 = 3(5 - 4) \). Calculate both sides to confirm they are equal: \(10 - 7 = 3 \times 1\), which results in \( 3 = 3 \). The solution is verified.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive property
The distributive property is crucial in solving equations like the one we're tackling here. It allows you to remove parentheses by distributing a multiplicative factor across terms inside those parentheses.
Let's break it down with our example equation: \( 2x - 7 = 3(x - 4) \). You see the term \( 3(x - 4) \) on the right-hand side.
Applying the distributive property means multiplying 3 by both \(x\) and \(-4\). Here's what happens:
  • Multiply 3 by \(x\) to get \(3x\).
  • Multiply 3 by \(-4\) to get \(-12\).
So, after applying the distributive property, we simplify our equation to \( 2x - 7 = 3x - 12 \). This step is vital as it sets the stage to solve the equation by isolating the variable.
Isolating the variable
After applying the distributive property, the next step in solving an equation is to isolate the variable, in this case, \(x\).
This means you'll want \(x\) on one side of the equation by itself. Let's follow the steps to do that.
From our previous equation \( 2x - 7 = 3x - 12 \), the variable terms \(2x\) and \(3x\) are not on the same side. So we need to move them. Here's how:
  • Subtract \(2x\) from both sides: \( 2x - 7 - 2x = 3x - 12 - 2x \).
  • This simplifies to \(-7 = x - 12\).
Next, we need to get \(x\) by itself. You can do this by eliminating the number with \(x\), which in this case is \(-12\).
Add 12 to both sides: \(-7 + 12 = x - 12 + 12\), which simplifies to \(5 = x\).
Now \(x\) is isolated, and the solution is clear: \(x = 5\).
Verify the solution
Once you've found a solution, it's important to verify it. Verifying ensures your solution is correct and satisfies the original equation.
We found \(x = 5\). Let's substitute this value back into the original equation: \(2x - 7 = 3(x - 4) \).
Replace \(x\) with 5: \(2(5) - 7 = 3(5 - 4)\). Now calculate both sides:
  • For the left side: \(2(5) - 7 = 10 - 7 = 3\).
  • For the right side: \(3(1) = 3\).
Since both sides equal 3, the solution \(x = 5\) is verified to be correct. Verifying adds confidence to your problem-solving process and ensures no errors were made.

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