Problem 73 - Algebra for College Students

Question

Solve each equation, and check the solution. \(0.05 x+0.12(x+5000)=940\)

Step-by-Step Solution

Answer

The solution is \(x = 2000\).

Step 1: Distribute the Constants

Distribute the 0.12 across the terms inside the parentheses: \(0.12(x + 5000) = 0.12x + 600\). This gives the equation: \(0.05x + 0.12x + 600 = 940\).

Step 2: Combine Like Terms

Combine the terms with \(x\): \(0.05x + 0.12x = 0.17x\). This updates our equation to: \(0.17x + 600 = 940\).

Step 3: Isolate the Variable

Subtract 600 from both sides of the equation to isolate the term with \(x\): \(0.17x = 340\).

Step 4: Solve for \(x\)

Divide both sides of the equation by 0.17 to solve for \(x\): \(x = \frac{340}{0.17} = 2000\). The solution is \(x = 2000\).

Step 5: Check the Solution

Substitute \(x = 2000\) back into the original equation to verify: \(0.05(2000) + 0.12(2000 + 5000) = 100 + 0.12(7000) = 100 + 840 = 940\). The solution checks out correctly.

Key Concepts

distributive property

The distributive property in algebra helps to simplify an equation by distributing multiplication over addition or subtraction within parentheses. In the equation provided, we have the term 0.12(x + 5000). To simplify this, you multiply 0.12 by each term inside the parentheses:

0.12(x) + 0.12(5000).

This gives us:

0.12x + 600.

Now the equation looks simpler and ready for the next steps: 0.05x + 0.12x + 600 = 940.

combining like terms

After using the distributive property, the next step is combining like terms. Like terms are terms that have the same variables raised to the same power. In our equation, we combined the terms that both have x:

0.05x and 0.12x.

When we add these coefficients together, we get:

0.05x + 0.12x = 0.17x.

So our equation becomes:

0.17x + 600 = 940.

This step simplifies our equation further and brings us closer to solving for x.

isolating variables

This step focuses on isolating the variable on one side of the equation to solve for it. In our simplified equation 0.17x + 600 = 940, we aim to isolate x. We do this by getting rid of the constant term (600) on the same side as the variable. Subtract 600 from both sides:

0.17x + 600 - 600 = 940 - 600.

This simplifies to:

0.17x = 340.

Now, x is almost isolated. The final step involves solving for x.

checking solutions

Once we find a solution, it's crucial to check it to ensure accuracy. In our case, we found that x = 2000. We substitute this value back into the original equation:

0.05(2000) + 0.12(2000 + 5000).

First, calculate each term:
0.05 * 2000 = 100
0.12 * 7000 = 840.

So, 100 + 840 = 940.

Since both sides of the equation are equal, our solution is verified as correct.