/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 97 Solve: \(2(x-9)=5 x-3(x+6)\)... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 97

Solve: \(2(x-9)=5 x-3(x+6)\)

Short Answer

Expert verified
The equation is true for all values of x.

Step by step solution

01

- Distribute on both sides

First, distribute the terms inside the parentheses:On the left side: 2(x-9) = 2x - 18.On the right side: 5x - 3(x + 6) = 5x -3x - 18.
02

- Simplify both sides

Simplify the expressions from the previous step: Left side remains: 2x - 18.Right side: 5x - 3x - 18 = 2x - 18.
03

- Set both sides equal and solve for x

Since both sides of the equation are equal, 2x - 18 = 2x - 18. There isn鈥檛 a specific value of x that satisfies the equation; it is true for all values of x.

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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 useful tool in algebra. It helps to eliminate parentheses by distributing a multiplier across the terms inside. For example, in the equation \( 2(x-9) \), the 2 is distributed to both \( x \) and \( -9 \). This simplifies to \( 2x - 18 \).
Similarly, for \( 5x - 3(x + 6) \), the -3 is distributed to both \( x \) and \( 6 \), which simplifies to \( 5x - 3x - 18 \).

It's important to remember to apply the distributive property to every term inside the parentheses to avoid mistakes.
Simplifying Expressions
Once you've applied the distributive property, the next step is simplifying the expressions. Simplifying involves combining like terms, which are terms that have the same variable raised to the same power.

For instance, after distributing on the right side of the given equation: \( 5x - 3(x + 6) \), we get \( 5x - 3x - 18 \).
Combine the like terms, which in this case are \( 5x \) and \( -3x \). This results in:\( 2x - 18 \).

By simplifying expressions, we make it easier to solve for the unknown variable.
Infinite Solutions
When both sides of a linear equation simplify to the same expression, it means that the equation holds true for any value of the variable. This situation is known as having infinite solutions.

In the provided example, after simplifying both sides of the equation \( 2(x-9)=5 x-3(x+6) \) we get:\( 2x - 18 = 2x - 18 \).
Both sides are identical, so the equation is always true, regardless of the value of \( x \).

Infinite solutions scenario is an important concept because it shows us that the relationship described by the equation is universally valid.

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Most popular questions from this chapter

Use a graphing calculator to graph each equation. Choose a window that shows the \(x\)-intercept and \(y\)-intercept. Sketch the graph; describe the window. \(y=2 x+5\)

Explain why the \(x\)-intercept, the \(y\)-intercept, and one other point are not three different solutions of \(y=2 x\).

The completed problem has one mistake. (a) Describe the mistake in words, or copy down the whole problem and highlight or circle the mistake. (b) Do the problem correctly. Problem: Write the equation of the vertical line that passes through \((6,3)\). Incorrect Answer: \(y=3\)

Use the slope formula to find the slope of the line that passes through the points. \((-6,-3) ;(-9,-2)\)

For exercises 91-94, use a graphing calculator to graph each equation. Choose a window that shows the \(x\)-intercept and \(y\)-intercept. Sketch the graph; describe the window. \(y=-2 x+5\)
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