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Chapter 2: Problem 28

Solve each linear equation. Show your work and check your answer. $$-3 x-4=11$$

Short Answer

Expert verified
x = -5

Step by step solution

01

Add 4 to both sides

To isolate the term with the variable, add 4 to both sides of the equation: -3x - 4 + 4 = 11 + 4 This simplifies to: -3x = 15
02

Divide by -3

Now, divide both sides of the equation by -3 to solve for x: \(-3x \div (-3) = 15 \div (-3)\) This simplifies to: x = -5
03

Check your solution

Substitute x = -5 back into the original equation to verify: -3(-5) - 4 = 11 This simplifies to: 15 - 4 = 11 11 = 11 The original equation holds true, so the solution x = -5 is correct.

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Key Concepts

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

Isolating Variables
Isolating the variable is a key step in solving linear equations. It means getting the variable (e.g., x) alone on one side of the equation. This helps you see exactly what the variable equals. In the original equation, \(-3x - 4 = 11\), we needed to isolate x.
Here's a step-by-step approach:
  • Step 1: Move any constants to the other side. In this example, we added 4 to both sides to cancel out the -4 attached to the \-3x. This gets us \-3x = 15.
  • Step 2: Next, divide both sides by the coefficient of the variable. Here, -3 is the coefficient of x, so we divide both sides by -3. This simplifies to x = -5.
Remember, the goal is to simplify the equation so you can clearly see the value of x. Always perform the same operation on both sides to keep the equation balanced.
Checking Solutions
After solving a linear equation, it's crucial to check your solution. This ensures you didn't make any calculation errors. Here's how to check your solution:
  • First, substitute the found value back into the original equation. In our example, we substitute x = -5 into \(-3x - 4 = 11\).
  • This turns the equation into \-3(-5) - 4 = 11 \ which simplifies to \15 - 4 = 11\.
  • Finally, confirm that both sides are equal. If they are (as they are in our example, since 11 = 11), then your solution is correct.
Checking solutions is like redoing your homework. Even if you are confident, this step catches any unnoticed mistakes.
Linear Equations
A linear equation is an equation that forms a straight line when graphed. It contains variables raised only to the first power, without any exponents, roots, or other operations. The general form is \Ax + B = C\ where A, B, and C are constants.
This implies that changes in one variable will cause proportional changes in the other. For instance, in \(-3x - 4 = 11\), the equation tells us how changes in x affect the left side of the equation.
Key characteristics of linear equations include:
  • Simplicity: They are straightforward and involve basic arithmetic operations.
  • Solutions: Solving simply means finding the value of the variable that makes the equation true.
  • Graphing: If plotted on a coordinate plane, they produce a straight line.
Understanding these foundational aspects will make more complex equations easier to tackle.

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