Problem 10 - Prealgebra

Question

Check to see if the number to the right of each of the following equations is the solution to the equation. $$x-8=3 x+2 ;-5$$

Step-by-Step Solution

Answer

Yes, \(-5\) is the solution to the equation since both sides equal \(-13\).

Step 1: Understand the Problem

We need to check if \(x = -5\) is a solution to the equation \(x - 8 = 3x + 2\). This means substituting \(x = -5\) into the equation and seeing if both sides are equal.

Step 2: Substitute the Value of x

Replace \(x\) with \(-5\) in the equation: \(-5 - 8 = 3(-5) + 2\).

Step 3: Calculate the Left Side

Calculate \(-5 - 8\). This simplifies to \(-13\).

Step 4: Calculate the Right Side

Calculate \(3(-5) + 2\). First, compute \(3 \times (-5)\), which equals \(-15\). Then add \(-15 + 2 = -13\).

Step 5: Compare Both Sides

Compare the results from the left and right sides: The left side is \(-13\) and the right side is also \(-13\). Since both sides are equal, \(x = -5\) is the solution to the equation.

Key Concepts

Solution Verification

After calculating both the left and right sides of the equation separately, it’s important to compare the results to verify the solution. Solution verification is the process of confirming whether a particular value satisfies a given equation. Here, our exercise asks us to check if the number \(x = -5\) solves the equation \[x - 8 = 3x + 2\]This involves substituting the proposed solution into the equation and then calculating both the left-hand side (LHS) and right-hand side (RHS). If both sides of the equation are equal, then the proposed number is indeed a solution.

For our problem:- Substitute \(x = -5\) into the equation.- Calculate each side:

• LHS: \(-5 - 8 = -13\)
• RHS: \(3(-5) + 2 = -13\)

Both the LHS and RHS equal to \(-13\), verifying that \(x = -5\) is a valid solution. It's important to check both sides to ensure no mistakes were made in substitution or calculation.

Substitution Method

The substitution method is a key mathematical technique used to find solutions to equations. It involves replacing the variable in an equation with a specific number to test if it satisfies the equation. In this exercise, we are using substitution to check if \(x = -5\) is a solution. By substituting the value:

• Replace \(x\) with \(-5\) in the equation \(x - 8 = 3x + 2\)
• This step transforms the equation into \(-5 - 8 = 3(-5) + 2\)

Substituting gives us concrete numbers to work with instead of variables. This allows us to directly perform arithmetic operations and verify the equality. Remember:- Use parentheses for clarity:

• For multiplication, write \(3(-5)\) instead of simply \(3 \times -5\).

This clarity helps prevent errors during substitution and simplifies arithmetic processes involved.

Equation Simplification

Simplifying an equation is a vital step when solving it or checking if a value satisfies it. The goal of simplification is to reduce the equation to its simplest form, making calculations straightforward. In this exercise, simplification happens on both sides of the equation:

• For the Left-Hand Side: Start with \(-5 - 8\)
• Compute it to get \(-13\)
• For the Right-Hand Side: Calculate \(3(-5) + 2\)
• First, solve \(3 \times (-5)\) which results in \(-15\)
• Then, add \(-15 + 2\), arriving at \(-13\)

This step-by-step breakdown of each side allows you to see how the equation behaves and better understand the solution. Simplification is essential because it ensures accuracy when verifying if the given value is a solution.