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Chapter 2: Problem 80

\(2(3 x-4)-5 x=7\)

Short Answer

Expert verified
x = 15

Step by step solution

01

- Distribute

First, distribute the 2 through the parentheses: \[ 2(3x - 4) - 5x = 7 \] becomes \[ 6x - 8 - 5x = 7 \].
02

- Combine like terms

Next, combine the like terms on the left-hand side: \[ 6x - 5x - 8 = 7 \] which simplifies to \[ x - 8 = 7 \].
03

- Isolate x

Add 8 to both sides to isolate \( x \): \[ x - 8 + 8 = 7 + 8 \] which simplifies to \[ x = 15 \].

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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 a fundamental concept in algebra that helps make equations easier to solve. It involves multiplying a single term by each term inside a set of parentheses. In our example, we start with the equation:

2(3x - 4) - 5x = 7

When we apply the distributive property, we multiply 2 by both 3x and -4. This gives:
2 * 3x + 2 * (-4) = 6x - 8

Now the equation looks like: 6x - 8 - 5x = 7.

The primary goal here is to eliminate the parentheses, allowing us to combine like terms more easily later on.
Combining Like Terms
Combining like terms simplifies the equation further by adding or subtracting terms that have the same variables. For example, '6x' and '-5x' are like terms because they both contain the variable 'x'.
In our equation: 6x - 8 - 5x = 7,

we combine '6x' and '-5x' to get: 6x - 5x = x.

So now our equation becomes: x - 8 = 7.

Combining like terms makes the equation simpler by reducing the number of terms.
Isolating Variables
The final step in solving our equation involves isolating the variable, which means getting 'x' by itself on one side of the equation.

Starting from: x - 8 = 7,
te we need to get rid of the '-8' next to 'x'.
To do this, we add 8 to both sides of the equation:
x - 8 + 8 = 7 + 8
x = 15.

By isolating 'x', we find that x equals 15. This process of isolating the variable helps us find the solution to algebraic equations.

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Most popular questions from this chapter

The product of \(8,\) and a number increased by \(6,\) is \(104 .\) What is the number?

In one day, a store sold \(\frac{2}{3}\) as many DVDs as Blu-ray discs. The total number of DVDs and Blu-ray discs sold that day was \(280 .\) How many DVDs were sold?

The difference of a number and 11 is \(-31 .\) What is the number?

The sum of the measures of the angles of any triangle is \(180^{\circ} .\) In triangle \(A B C,\) the measure of angle \(A\) is \(141^{\circ}\) greater than the measure of angle \(B\). The measure of angle \(B\) is the same as the measure of angle \(C\). Find the measure of each angle.

Graph the solution set, and write it using interval notation \(1 \leq 3+\frac{2}{3} p \leq 7\)
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