Problem 44 - Algebra A Combined Function

Question

Solve. $$ 15 x+20-10 x-9=25 x+8-21 x-7 $$

Step-by-Step Solution

Answer

The solution is \(x = -10\).

Step 1: Combine Like Terms

First, we need to simplify both sides of the equation by combining like terms. On the left side of the equation, combine the terms with \(x\): \(15x - 10x\), resulting in \(5x\). For the constant terms, combine \(20 - 9\) to get \(11\). Thus, the left side becomes \(5x + 11\). On the right side, combine \(25x - 21x\) to yield \(4x\), and combine the constant terms \(8 - 7\) to get \(1\). Therefore, the right side becomes \(4x + 1\). Now the equation looks like: \(5x + 11 = 4x + 1\).

Step 2: Isolate the Variable 'x'

Next, we aim to solve for \(x\) by getting all terms with \(x\) on one side and constants on the other. Subtract \(4x\) from both sides to get: \(5x - 4x + 11 = 1\), which simplifies to \(x + 11 = 1\).

Step 3: Solve for 'x'

Subtract 11 from both sides to isolate \(x\): \(x = 1 - 11\), giving \(x = -10\).

Step 4: Verify the Solution

Substitute \(x = -10\) back into the original equation to ensure both sides equate. The left side becomes \(15(-10) + 20 - 10(-10) - 9\) which simplifies to \(-150 + 20 + 100 - 9 = -39\). The right side becomes \(25(-10) + 8 - 21(-10) - 7\) which simplifies to \(-250 + 8 + 210 - 7 = -39\). The original equation holds true, confirming \(x = -10\) is correct.

Key Concepts

Combining Like Terms

When solving linear equations, one of the initial steps is simplifying the equation by combining like terms. Like terms contain the same variable raised to the same power. In our example, the variable is \( x \), and they appear both on the left and the right side of the equation. You start by grouping these similar terms.

• On the left side, combine the terms \( 15x - 10x \), which gives you \( 5x \).
• Separate the constant terms too: \( 20 - 9 \) becomes \( 11 \).

Hence, the left side of the equation simplifies to \( 5x + 11 \).Similarly, for the right side, combine the like terms:

• The \( x \) terms \( 25x - 21x \) simplify to \( 4x \).
• The constant terms \( 8 - 7 \) become \( 1 \).

Therefore, the right side of the equation simplifies to \( 4x + 1 \).After combining like terms, the simplified equation becomes \( 5x + 11 = 4x + 1 \). By reducing each side to its simplest form, this makes it easier to proceed to the next step - isolating the variable.

Isolate the Variable

Once you've combined like terms, your goal is to solve for the variable \( x \). This involves moving all the terms with \( x \) to one side of the equation and placing constants on the opposite side.Start by subtracting \( 4x \) from both sides of the equation. This operation will help you eliminate \( x \) terms from the right side, yielding:\[ 5x - 4x + 11 = 1 \] This simplifies to \( x + 11 = 1 \), making it clear that you are one step closer to isolating \( x \).Next, remove the constant on the side with \( x \) by subtracting 11 from both sides:\[ x = 1 - 11 \] This gives \( x = -10 \). By isolating \( x \), you've found a potential solution to the equation. Remember, isolating means keeping \( x \) alone on one side, ensuring we clearly identify its value.

Verifying Solutions

Verifying a solution in mathematics is important to confirm you've solved the equation correctly. By substituting the solved value back into the original equation, you can check if both sides are equal.Let's substitute \( x = -10 \) back into the original equation:1. **Substitute in the left side:** \[ 15(-10) + 20 - 10(-10) - 9 \] Simplifies to: \[ -150 + 20 + 100 - 9 = -39 \] 2. **Substitute in the right side:** \[ 25(-10) + 8 - 21(-10) - 7 \] Simplifies to: \[ -250 + 8 + 210 - 7 = -39 \]Both sides of the equation equal \( -39 \), confirming that \( x = -10 \) is indeed the correct solution. Verifying solutions ensures accuracy, helping avoid errors, and strengthening your understanding of how to solve linear equations.