Problem 29 - Elementary Algebra Within Reach

Question

Solve the equation and check your solution. $$-5 x-15=0$$

Step-by-Step Solution

Answer

The solution to the equation is x = -3.

Step 1: Addition to remove the number on the variable's side

The equation given is -5x - 15 = 0. We need the term with the x on one side of the equation hence we add 15 to both sides of the equation to balance out the -15 on the left. This results in -5x = 15.

Step 2: Division to isolate x

Now we have -5x = 15. To isolate x, we divide both sides by -5. Therefore, x = 15 / -5.

Step 3: Simplification

Simplify x = 15 / -5 to find that x = -3.

Step 4: Verification

Verify the found value for x by substituting it back into the original equation. So, -5(-3) - 15 = 15 - 15 = 0, which checks out.

Key Concepts

Understanding Linear Equations

Linear equations are mathematical expressions that represent a straight line when graphed. They have one or multiple variables, but each variable is only to the first power and not multiplied or divided by other variables. The standard form of a linear equation is represented as ax + b = c, where a, b, and c are constants, and x represents the variable.

The exercise -5x - 15 = 0 is a simple linear equation with one variable (x). The aim here is to find the value of x that makes the equation true. Dealing with linear equations often involves basic operations such as addition, subtraction, multiplication, and division to simplify the equation and eventually find the value of the variable involved.

Isolating Variables

Isolating the variable is an essential technique in solving linear equations as it helps to find the value of an unknown. To isolate the variable, you perform inverse operations to cancel out other numbers or terms in the equation. The goal is to get the variable by itself on one side of the equation, equaling a number or a simpler expression on the other side.

In the equation -5x - 15 = 0, we first add 15 to both sides to cancel out the -15.

Addition to Isolate

-5x - 15 + 15 = 0 + 15 simplifies to -5x = 15. Next, to get x by itself, we divide both sides by -5.

Division to Find the Variable

-5x / -5 = 15 / -5, giving us x = -3.

Verification of Solutions

Once you have a proposed solution to a linear equation, it is crucial to verify that it is correct. This step ensures that the value you found for the variable indeed satisfies the original equation. Verification involves substituting the value back into the original equation and seeing if the equation balances.

For the solution x = -3, we verify by replacing x in the equation -5x - 15 = 0.

Substitution for Verification

We compute -5(-3) - 15 and simplify to 15 - 15, which equals 0. Since this matches the original equation, our solution x = -3 is verified. This final check is a vital step—it's like proofreading your work in a math problem!