Problem 17 - Algebra 1: Concepts and Skills

Question

Solve the equation. Check your solution in the original equation. $$ \frac{3}{8}(16 x-8)=9-5(x-2) $$

Step-by-Step Solution

Answer

Solution for \(x\) in the equation \(\frac{3}{8}(16 x-8)=9-5(x-2)\) is \(x=2\).

Step 1: Distribute the fractions and simplify

We start by applying the property of distribution, to get rid of the brackets: It gives us the equation \(6x-3=9-5x+10\). After simplifying this equation, it is transformed to \(6x-3=19-5x\).

Step 2: isolate the variable \(x\)

Next, we combine the \(x\) terms on one side and the constants on the other side of the equation by adding \(5x\) to both sides and adding 3 to both sides, which gives us \(11x=22\). Now, to isolate \(x\), we divide both sides by 11, resulting in \(x=2\) as the solution of this equation.

Step 3: Verify the solution

We substitute the value of \(x=2\) in the original equation to verify it. If both sides of the equation have the same value then our solution is correct. After substitution, we get \(\frac{3}{8}(16(2)-8)=9-5((2)-2)\), which simplifies to \(3=3\), thus confirming the correctness of the solution.

Key Concepts

Distribution Property in Algebra

The distribution property in algebra is a fundamental tool used to simplify equations by removing parentheses and distributing terms across expressions. In the given problem, we start with the equation \( \frac{3}{8}(16x-8)=9-5(x-2) \). Here we need to distribute the terms inside the parentheses to proceed with solving the equation.

To apply this property effectively, we multiply each term inside the parentheses by the outside expression. This means for the left side \( \frac{3}{8} \times 16x \) and \( \frac{3}{8} \times -8 \). The right side requires distributing \(-5\) over \(x-2\), so we multiply \(-5\) by both \(x\) and \(-2\).

• Left Side Distribution: \( \frac{3}{8} \times 16x = 6x \) and \( \frac{3}{8} \times -8 = -3 \)
• Right Side Distribution: \(-5 \times x = -5x \) and \(-5 \times -2 = +10 \)

After distributing, the equation becomes \( 6x - 3 = 9 - 5x + 10 \). Simplifying further involves combining like terms, particularly the constants \(-3\) and \(+10\), resulting in the simplified equation \( 6x - 3 = 19 - 5x \).

Isolating the Variable

Once we have a simplified equation, the next step is isolating the variable, typically represented as \(x\). This step is crucial to finding the value of \(x\), which is the solution to our equation.

In the simplified equation \(6x - 3 = 19 - 5x\), our goal is to get \(x\) alone on one side. To do this, follow these steps:

• Add \(5x\) to both sides to move \(x\) terms to the same side: \(6x + 5x - 3 = 19\)
• This simplifies to \(11x - 3 = 19\)
• Next, add \(3\) to both sides to shift constants to the right: \(11x = 22\)
• Finally, divide both sides by \(11\) to solve for \(x\): \(x = 2\)

By isolating the variable, you determine \(x = 2\). This result represents the solution to our original equation. Ensuring each step involves balancing both sides is the key to successfully isolating the variable.

Verifying Solutions in Equations

After solving an equation, verifying the solution is a vital step. It ensures that the solution is correct and satisfies the original equation's conditions. To verify, we substitute the calculated value back into the original equation.

For our example, substitute \(x = 2\) into the original equation \(\frac{3}{8}(16x-8)=9-5(x-2)\). This means recalculating both sides with \(x = 2\) to ensure they are equal:

• Substitute into the left side: \(\frac{3}{8}(16(2)-8) = \frac{3}{8}(32-8) = \frac{3}{8} \times 24 = 3\)
• Substitute into the right side: \(9-5((2)-2) = 9-5 \times 0 = 9\)

Since both sides equal \(3\), the solution \(x = 2\) satisfies the original equation.

The verification process is simple but essential. It acts as a double-check, ensuring no computational errors during solving and confirming the solution's validity.