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Chapter 2: Problem 35

Solve each linear equation. Show your work and check your answer. $$2(x+9)-x=36$$

Short Answer

Expert verified
x = 18

Step by step solution

01

Distribute

Multiply out the parentheses: $$2(x + 9) - x = 36$$ Expand it to: $$2x + 18 - x = 36$$
02

Combine Like Terms

Combine the terms involving x: $$2x - x + 18 = 36$$ Simplify it to: $$x + 18 = 36$$
03

Isolate the Variable

Subtract 18 from both sides to get the variable alone: $$x + 18 - 18 = 36 - 18$$ This simplifies to: $$x = 18$$
04

Check Your Answer

Substitute the value of x back into the original equation to ensure it satisfies the equation: $$2(x + 9) - x = 36$$ Substitute x = 18: $$2(18 + 9) - 18 = 36$$ Simplify it to: $$2 \times 27 - 18 = 36$$ Which further simplifies: $$54 - 18 = 36$$ Finally: $$36 = 36$$ The solution is correct.

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Key Concepts

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

Distributive Property
When solving linear equations, the distributive property allows you to remove parentheses by distributing a multiplier to each term inside. For example, in the equation \2(x+9)-x=36\, we first apply the distributive property: \(2(x + 9)\).
This means you multiply both parts inside the parentheses by 2, leading to \(2 \times x + 2 \times 9\).
So, \(2(x + 9)\) becomes \(2x + 18\).
This step simplifies the expression and makes it easier to combine like terms.
Combining Like Terms
Combining like terms means adding or subtracting terms that have the same variable part. After using the distributive property, the equation \(2x + 18 - x = 36\) has terms that can be combined.
Notice that both \(2x\) and \(-x\) are like terms since they both contain the variable \(x\).
Summing these terms results in \(2x - x\), which simplifies to \(x\).
Thus, the equation \(2x + 18 - x = 36\) simplifies to \(x + 18 = 36\).
This step reduces the complexity of the equation.
Isolating the Variable
To isolate the variable means to get the variable \(x\) alone on one side of the equation. From the previous step, you've simplified the equation to \(x + 18 = 36\).
You then subtract 18 from both sides to isolate \(x\).
Performing this operation gives: \(x + 18 - 18 = 36 - 18\), which simplifies to \(x = 18\).
This step is crucial because it reveals the value of the unknown variable.
Checking Solutions
Checking your solution verifies that your answer is correct. Substitute the found value back into the original equation: \(2(x + 9) - x = 36\), with \(x = 18\).
Substituting gives: \(2(18 + 9) - 18 = 36\).
Simplify inside the parentheses: \(2 \times 27 - 18 = 36\).
Then multiply: \(54 - 18 = 36\).
Finally confirm: \(36 = 36\).
Since both sides of the equation are equal, the solution \(x = 18\) is correct and satisfies the original equation.

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