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Chapter 8: Problem 136

Solve the equations for the variable. $$6 x-15=5 x+3$$

Short Answer

Expert verified
x = 18

Step by step solution

01

- Isolate variable terms on one side

First, subtract \(5x\) from both sides of the equation to isolate the variable terms on one side. \[6x - 5x - 15 = 5x - 5x + 3\]This simplifies to:\[x - 15 = 3\]
02

- Isolate the variable

Next, add 15 to both sides of the equation to isolate the variable. \[x - 15 + 15 = 3 + 15\]This simplifies to:\[x = 18\]

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Key Concepts

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

Isolate Variable
In solving linear equations, one key step is to isolate the variable. This means you want the variable (like x) to be alone on one side of the equation.

For example, in the equation \(6x - 15 = 5x + 3\), the goal is to have 'x' on one side by itself. To do this, you need to move all other terms to the opposite side.

Start by subtracting \(5x\) from both sides:
\[6x - 5x - 15 = 5x - 5x + 3\]

This simplifies to:
\[x - 15 = 3\].

Now, add 15 to both sides to get 'x' by itself:
\[x - 15 + 15 = 3 + 15\]

This simplifies to:
\[x = 18\]

By performing these operations, you've successfully isolated the variable 'x'.
Linear Equations
Linear equations are algebraic expressions that represent straight lines when graphed. They have the general form:
\[ax + b = c\]
where 'a', 'b', and 'c' are constants.

In the exercise, we had:
\[6x - 15 = 5x + 3\]
This is a linear equation because it follows the same format.

When solving linear equations:
  • Always aim to isolate the variable (x).
  • Perform the same operation on both sides to keep the equation balanced.
  • Simplify step-by-step to make calculations easier.
Understanding linear equations is crucial, as they form the basis for more complex algebra. Once you master this, you鈥檒l find it easier to move on to quadratic equations and beyond.
Step-by-Step Algebra
Solving linear equations using a step-by-step approach helps ensure accuracy and clarity. Let鈥檚 break down the process again using the example \(6x - 15 = 5x + 3\):

Step 1: Isolate variable terms by subtracting \(5x\) from both sides:
\[6x - 5x - 15 = 5x - 5x + 3\]

This gives:
\[x - 15 = 3\]

Step 2: Isolate 'x' by adding 15 to both sides:
\[x - 15 + 15 = 3 + 15\]

This simplifies to:
\[x = 18\]

Key points for step-by-step algebra:
  • Focus on isolating the variable term first.
  • Perform inverse operations (e.g., subtract if you have addition).
  • Keep the equation balanced by applying changes to both sides.
  • Simplify gradually鈥攄on鈥檛 try to do too much at once.
Following these steps will help you master linear equations and build a strong foundation in algebra.

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Most popular questions from this chapter

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. $$ 0.45 x=6.75 $$

In the following exercises, solve each equation. $$ x-15=-42 $$

In the following exercises, solve each equation. $$ 0.1 d+0.25(d+8)=4.1 $$

In the following exercises, solve the equations with constants on both sides. $$ 8 p+7=47 $$
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