Problem 76 - Algebra for College Students

Question

Solve each equation, and check the solution. \(0.04(90)+0.12 x=0.06(460+x)\)

Step-by-Step Solution

Answer

x = 400

Step 1 - Distribute the constants

Distribute the constants to both terms inside the parentheses. This gives: \[0.04 \times 90 + 0.12 x = 0.06 \times 460 + 0.06 x\]

Step 2 - Calculate the multiplication

Perform the multiplications on both sides of the equation. \[3.6 + 0.12 x = 27.6 + 0.06 x\]

Step 3 - Move all x terms to one side

Subtract \(0.06x\) from both sides to get all the \(x\) terms on one side of the equation. \[3.6 + 0.06 x = 27.6\]

Step 4 - Isolate the variable

Subtract 3.6 from both sides to isolate the variable term. \[0.06 x = 24\]

Step 5 - Solve for x

Divide both sides by 0.06 to solve for \(x\). \[ x = \frac{24}{0.06} = 400\]

Step 6 - Check the solution

Substitute \(x = 400\) back into the original equation to check: \[0.04(90) + 0.12(400) = 0.06(460 + 400)\] Calculate both sides: \[3.6 + 48 = 0.06 \times 860\] \[51.6 = 51.6\] Since both sides are equal, \(x = 400\) is correct.

Key Concepts

distribute constants

Distributing constants is the first step we take when solving linear equations like this one. It means we multiply everything inside the parentheses by the number outside. In our exercise, we have the equation: \(0.04 (90) + 0.12x = 0.06 (460 + x)\).Let's break it down further:

• First, multiply \( 0.04 \) by 90, giving us \( 3.6 \).
• Then, distribute \( 0.06 \) to both 460 and \( x \) inside the parentheses, so \( 0.06 \times 460 \) becomes \( 27.6 \) and \( 0.06 \times x \) remains \( 0.06x \).

This gives us the new simplified equation: \[3.6 + 0.12x = 27.6 + 0.06x\]. Remember, distributing constants helps us eliminate the parentheses and makes the equation easier to solve step by step.

isolating variables

After distributing the constants, our next goal is to isolate the variable (in this case, \( x \)) on one side of the equation. We use basic algebraic operations to achieve this. In the simplified equation \[3.6 + 0.12x = 27.6 + 0.06x\], we want all terms with \( x \) on one side. Here's how:

• Subtract \( 0.06x \) from both sides. This leaves us with \(3.6 + 0.06x = 27.6\).

Now, we have fewer \( x \) terms to deal with. Next, we isolate the \( x \) term by moving constants to the opposite side:

• Subtract 3.6 from both sides, giving us \( 0.06x = 24 \).

Finally, to solve for \( x \), divide both sides by 0.06, resulting in \( x = 400 \). Isolating variables is like peeling layers of an onion, removing one layer at a time until only the variable is left.

checking solutions

Once we've found our solution, it's essential to check its correctness by substituting it back into the original equation. In our problem, we found \( x = 400 \). Let's substitute this value into the original equation \[0.04 (90) + 0.12 (400) = 0.06 (460 + 400)\]. Calculate both sides step-by-step to ensure they are equal:

• First, on the left side: \( 0.04 \times 90 + 0.12 \times 400 = 3.6 + 48 = 51.6 \).
• Next, on the right side: \( 0.06 \times (460 + 400) = 0.06 \times 860 = 51.6 \).

Since both sides equal 51.6, our solution \( x = 400 \) is indeed correct. Checking solutions confirms that we haven't made any mistakes, ensuring the accuracy of our final answer.