Problem 13 - Algebra A Combined Function

Question

Solve each equation. Don't forget to first simplify each side of the equation, if possible. Check each solution. See Examples 5 through 7 . $$ 7 x+2 x=8 x-3 $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = -3\).

Step 1: Combine Like Terms

On the left side of the equation, combine the terms with the variable \(x\). Simplify \(7x + 2x\) to get \(9x\). The equation is now: \(9x = 8x - 3\).

Step 2: Isolate Variable

To isolate \(x\), subtract \(8x\) from both sides of the equation. This gives \(9x - 8x = -3\), simplifying to \(x = -3\).

Step 3: Verify Solution

Substitute \(x = -3\) back into the original equation to check the solution: \(7(-3) + 2(-3) = 8(-3) - 3\). This translates to: \(-21 - 6 = -24 - 3\). Simplifying both sides gives \(-27 = -27\), confirming that the solution is correct.

Key Concepts

Combining Like Terms

Understanding how to combine like terms is an essential step in solving linear equations. Like terms are terms in an equation that have the same variable raised to the same power. To simplify an equation, you need to combine these terms by either adding or subtracting them. For example, in the equation given in the exercise, we have terms involving the variable \(x\) on the left side: \(7x + 2x\). Since both of these terms have the same variable \(x\), you can combine them by adding their coefficients together. This will give you \(9x\). This simplification makes it much easier to manipulate and solve the equation.
Combining like terms is often the first move you make when tackling an equation. It helps manage complexity and sets up the equation for the next steps, such as isolating the variable.

Isolating Variables

Once like terms are combined, the next step is to isolate the variable—typically \(x\)—on one side of the equation. This means moving all instances of the variable to one side, leaving the constants on the other. In the step-by-step solution, the equation after combining like terms was \(9x = 8x - 3\).
To isolate \(x\), you need to eliminate the \(8x\) from the right side by subtracting \(8x\) from both sides. This results in \(9x - 8x = -3\), simplifying to \(x = -3\).
Isolating the variable is necessary because it allows you to solve for the variable directly. You essentially restructure the equation to easily identify the value of the variable in question. If done correctly, you will have your solution, ready to be verified.

Checking Solutions

After isolating the variable and finding your solution, it's critical to check whether your solution satisfies the original equation. This step acts as a verification method to ensure no errors were made in your calculations.
For our example, substitute \(x = -3\) back into the original equation: \(7(-3) + 2(-3) = 8(-3) - 3\). Calculate each term:

• \(7(-3) = -21\)
• \(2(-3) = -6\)
• \(8(-3) = -24\)

Combine the terms on both sides: \(-21 - 6 = -24 - 3\) simplifies to \(-27 = -27\).
Since both sides of the equation are equal, this confirms that the solution \(x = -3\) is correct. Performing this check ensures accuracy and builds confidence in the solution.