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Chapter 4: Problem 7

Solve each equation by doing the same thing to both sides. $$ 2 x+3=x+5 $$

Short Answer

Expert verified
x = 2

Step by step solution

01

Subtract x from both sides

Subtract x from both sides to get the x terms on one side. Starting with: \(2x + 3 = x + 5\) After subtracting x from both sides: \(2x - x + 3 = x - x + 5\) Which simplifies to: \(x + 3 = 5\)
02

Subtract 3 from both sides

Next, subtract 3 from both sides to isolate the variable x. Starting with: \(x + 3 = 5\) After subtracting 3 from both sides: \(x + 3 - 3 = 5 - 3\) Which simplifies to: \(x = 2\)

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Key Concepts

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

algebraic manipulation
Algebraic manipulation involves rearranging and simplifying algebraic expressions and equations. This is a fundamental skill in solving equations.
To solve equations like \[ 2x + 3 = x + 5 \], we perform a series of steps that follow the principles of algebra.
The goal is to perform operations that ensure the equation remains balanced while simplifying it to find the variable's value.
In our exercise, we started by subtracting x from both sides because this helps to consolidate the variable terms onto one side of the equation.
isolating variables
Isolating variables is a key step in solving any algebraic equation.
Our goal is to get the variable (x in this case) by itself on one side of the equation.
In the example \[ 2x + 3 = x + 5 \], we subtracted x from both sides first to simplify the terms involving x.
This gave us \[ x + 3 = 5 \].
Next, we subtracted 3 from both sides to completely isolate x.
Hence, \[ x = 2 \].
By isolating the variable, we make it straightforward to see the solution.
simplifying equations
Simplifying equations is essential for making them easier to solve.
It involves combining like terms and performing arithmetic operations to reduce the equation to its simplest form.
In our example, we first simplified \[ 2x + 3 = x + 5 \] by subtracting x from both sides, leading us to \[ x + 3 = 5 \].
Then, we simplified further by subtracting 3 from both sides, which resulted in \[ x = 2 \].
These simplifications are critical for solving the equation efficiently and accurately.

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Most popular questions from this chapter

Each table represents a linear relationship. Tell whether each relationship is increasing or decreasing. $$ \begin{array}{|c|c|}\hline x & {y} \\ \hline 1 & {3.2} \\ {-4} & {2.2} \\\ {0} & {3} \\ {-1} & {2.8} \\ {-3} & {2.4} \\ {-2} & {2.6} \\\ \hline\end{array} $$

Rewrite each expression as simply as you can. $$\left(a^{m}\right)^{n} \cdot\left(b^{3}\right)^{0}$$

Solve this system of quadratic equations by drawing a graph. $$ y=x^{2} \quad y=4-3 x^{2} $$

Graph the solution of each inequality on a number line. $$-\frac{3 p}{4}<6$$

Graph the solution of each inequality on a number line. $$12-5 q \geq 32$$
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