Problem 63 - Algebra for College Students

Question

Solve each equation and express the solutions in decimal form. Be sure to check your solutions. Use your calculator whenever it seems helpful. \(0.5(3 x+0.7)=20.6\)

Step-by-Step Solution

Answer

The solution is x = 13.5.

Step 1: Distribute the 0.5

Multiply 0.5 with each term inside the parentheses. This gives us:\[0.5(3x) + 0.5(0.7) = 20.6\]Simplifying this gives:\[1.5x + 0.35 = 20.6\]

Step 2: Isolate the term with x

Subtract 0.35 from both sides of the equation to isolate the term with x. This gives:\[1.5x = 20.6 - 0.35\]\[1.5x = 20.25\]

Step 3: Solve for x

Divide both sides by 1.5 to solve for x:\[x = \frac{20.25}{1.5}\]Calculate it using a calculator:\[x = 13.5\]

Step 4: Verify the solution

Substitute x = 13.5 back into the original equation to check:\[0.5(3(13.5) + 0.7) = 20.6\]\[0.5(40.5 + 0.7) = 20.6\]\[0.5(41.2) = 20.6\]\[20.6 = 20.6\]The original equation holds true when x = 13.5.

Key Concepts

Equation Verification

When solving linear equations, verifying your solution is crucial. It ensures that your answer is correct and that no mistakes were made during the process. Verification involves substituting the solution back into the original equation to check if it satisfies the equation. Let's see how this works through an example:

• After solving the equation, we found that \( x = 13.5 \).
• We substitute \( x = 13.5 \) back into the original equation \( 0.5(3x + 0.7) = 20.6 \).
• When we replace \( x \) with \( 13.5 \), we calculate: \( 0.5(3(13.5) + 0.7) = 20.6 \).
• The left side simplifies to \( 20.6 \), which is equal to the right side of the equation. This confirms that our solution is correct.

Verification helps catch any arithmetic or procedural errors. Always make it a habit to check your solutions.

Decimal Solutions

Working with decimal solutions can sometimes seem tricky, but it's just a matter of following the steps carefully. Decimals are numbers that have a whole part and a fractional part, expressed after a dot.Here's how we handle decimals in our equation:

• The equation given was \( 0.5(3x+0.7) = 20.6 \).
• By distributing, we got decimals in our expression: \( 1.5x + 0.35 \).
• When isolating \( x \), we performed operations that resulted in \( 1.5x = 20.25 \).
• Finally, dividing \( 20.25 \) by \( 1.5 \) gave us the decimal solution \( x = 13.5 \).

Using a calculator is recommended when dealing with decimals to avoid small calculation errors. As long as you perform the steps methodically, you'll handle decimals with confidence.

Distributive Property

The distributive property is a fundamental algebraic principle used to simplify expressions and solve equations. It states that \( a(b + c) = ab + ac \), which allows you to multiply a single term across terms inside parentheses.For our problem:

• The original equation was \( 0.5(3x + 0.7) = 20.6 \).
• Using the distributive property, 0.5 is multiplied by both \( 3x \) and \( 0.7 \).
• This yields \( 0.5 \times 3x = 1.5x \) and \( 0.5 \times 0.7 = 0.35 \).
• The new equation becomes \( 1.5x + 0.35 = 20.6 \).

By understanding and applying the distributive property correctly, you simplify the equation and make it solvable in fewer steps. Remember to apply it any time you see a term multiplied by a sum inside parentheses.