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Chapter 2: Problem 20

Solve each linear equation. Show your work and check your answer. $$3 x+5=26$$

Short Answer

Expert verified
x = 7

Step by step solution

01

- Subtract 5 from both sides

To isolate the term with the variable, subtract 5 from both sides of the equation:\[ 3x + 5 - 5 = 26 - 5 \]This simplifies to:\[ 3x = 21 \]
02

- Divide both sides by 3

Now, divide both sides of the equation by 3 to solve for \(x\):\[ \frac{3x}{3} = \frac{21}{3} \]This simplifies to:\[ x = 7 \]
03

- Check the solution

Substitute \(x = 7\) back into the original equation to verify the solution:\[ 3(7) + 5 = 26 \]Calculate the left side:\[ 21 + 5 = 26 \]Since both sides of the equation are equal, \(x = 7\) is correct.

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Key Concepts

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

Isolate the variable
When solving linear equations, our goal is to find the value of the variable (in this case, \(x\)). To do this, we need to isolate the variable on one side of the equation.
We proceed by performing operations that move other terms (constants and coefficients) away from the variable.
In our example, we start from the equation:

\(3x + 5 = 26\)
Here, the variable term is \(3x\) and the constant term is \(5\).
Before isolating \(x\), we first need to deal with the constant term.
Subtraction property of equality
To isolate the variable, we need to remove the constant term from the left side of the equation. We do this by using the subtraction property of equality, which states that if you subtract the same number from both sides of an equation, the equality remains balanced.
So, in our example, we subtract 5 from both sides:

\(3x + 5 - 5 = 26 - 5\)
This simplifies to:

\(3x = 21\)
By removing the constant term, we've successfully isolated the term with the variable.
Division property of equality
Next, we need to isolate the variable completely. The term with the variable is currently multiplied by 3. To get \(x\) by itself, we use the division property of equality. This property states that if you divide both sides of the equation by the same non-zero number, the equality remains balanced.
We divide both sides of the equation by 3:

\(\frac{3x}{3} = \frac{21}{3}\)
This simplifies to:
\(x = 7\)

Now, we've fully isolated \(x\) and found its value.
Verifying solutions
After solving the equation, we should always verify our solution to ensure it's correct. We do this by substituting the found value back into the original equation and checking if both sides are equal.
For our example, substitute \(x = 7\) back into the original equation:

\(3(7) + 5 = 26\)
Calculate the left side:

\(21 + 5 = 26\)
Since both sides of the equation are equal, we confirm that \(x = 7\) is indeed the correct solution.

Verification is an essential step because it confirms the accuracy of our solution and ensures that no mistakes were made during the solving process.

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