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Chapter 8: Problem 193

Solve the linear equation using the general strategy. $$5(x-4)-4 x=14$$

Short Answer

Expert verified
x = 34

Step by step solution

01

- Distribute the Constant

Distribute the constant 5 to both terms inside the parenthesis: \[ 5(x-4) = 5x - 20 \]So the equation becomes:\[ 5x - 20 - 4x = 14 \]
02

- Combine Like Terms

Combine the like terms on the left side of the equation:\[ 5x - 4x - 20 = 14 \]Simplify it to:\[ x - 20 = 14 \]
03

- Isolate the Variable

To isolate the variable x, add 20 to both sides of the equation:\[ x - 20 + 20 = 14 + 20 \]This simplifies to:\[ x = 34 \]

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Key Concepts

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

Distribute the Constant
When solving linear equations, one essential step is to handle parentheses effectively. We start by distributing the constant (or number) that is outside the parentheses to each term inside the parentheses. Let's take the equation from our exercise: \[5(x-4)-4x=14\] By distributing the constant 5 to both \( x \) and \( -4 \), we get: \[ 5(x-4) = 5x - 20 \] This simplifies our equation to: \[ 5x - 20 - 4x = 14 \] Now, the equation is easier to handle without parentheses.
Combine Like Terms
Next, we look to combine like terms. 'Like terms' are terms that have the same variables raised to the same power. In our simplified equation: \[ 5x - 20 - 4x = 14 \] The terms \( 5x \) and \( -4x \) are like terms because they both have the variable \( x \). By combining them: \[ 5x - 4x = x \] So our equation becomes: \[ x - 20 = 14 \] Now we have an even simpler equation to solve.
Isolate the Variable
The final step to solving the equation is isolating the variable, which means getting the variable by itself on one side of the equation. For the equation: \[ x - 20 = 14 \] we need \( x \) alone on the left side. We can achieve this by adding 20 to both sides of the equation: \[ x - 20 + 20 = 14 + 20 \] Simplifying the above, we get: \[ x = 34 \] This is our solution. By isolating the variable, we successfully determine that \( x = 34 \).

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Most popular questions from this chapter

In the following exercises, solve the equations with constants on both sides. $$ 10 w-5=65 $$

In the following exercises, solve the equations with constants and variables on both sides. $$ \frac{5}{8} c-4=\frac{3}{8} c+4 $$

Construction Miguel wants to drill a hole for a \(\frac{5}{8}\) -inch screw. The screw should be \(\frac{1}{12}\) inch larger than the hole. Let \(d\) equal the size of the hole he should drill. Solve the equation \(d+\frac{1}{12}=\frac{5}{8}\) to see what size the hole should be.

In the following exercises, solve each equation by clearing the decimals. $$ 0.6 p-1.9=0.78 p+1.7 $$

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