Problem 79 - Algebra for College Students

Question

Solve each equation, and check the solution. \(0.006(x+2)=0.007 x+0.009\)

Step-by-Step Solution

Answer

x = 3

Step 1 - Distribute the Coefficient

Distribute the coefficient 0.006 to both terms inside the parentheses: \(0.006(x + 2) = 0.006x + 0.012\). The equation becomes \(0.006x + 0.012 = 0.007x + 0.009\).

Step 2 - Move Terms Involving x to One Side

To isolate \(x\), subtract \(0.006x\) from both sides: \(0.012 = 0.007x - 0.006x + 0.009\). Simplify to \(0.012 = 0.001x + 0.009\).

Step 3 - Isolate x

Subtract 0.009 from both sides to isolate the term with \(x\): \(0.012 - 0.009 = 0.001x\). Simplify to \(0.003 = 0.001x\).

Step 4 - Solve for x

Divide both sides by 0.001 to solve for \(x\): \(\frac{0.003}{0.001} = x\). Simplify to \(x = 3\).

Step 5 - Check the Solution

Substitute \(x = 3\) back into the original equation to verify: \(0.006(3 + 2) = 0.007(3) + 0.009\). Simplify both sides: \(0.006(5) = 0.021 + 0.009\) which simplifies to \(0.03 = 0.03\). Since both sides are equal, \(x = 3\) is the correct solution.

Key Concepts

distributive property

The first step in solving the equation involves using the distributive property. This means you need to multiply a single term outside the parentheses by each term inside the parentheses.

In our example, we had the equation: \(0.006(x + 2)\).
By applying the distributive property, we get:
\(0.006(x) + 0.006(2)\).

This simplifies to:
\(0.006x + 0.012\).

It's important to do the multiplication step by step to avoid mistakes. Doing this correctly will make solving the rest of the equation straightforward.

isolating variables

Next, we need to isolate the variable. This means getting all terms with \(x\) on one side of the equation and the constant terms on the other side.

Starting from:
\(0.006x + 0.012 = 0.007x + 0.009\),
we subtract \(0.006x\) from both sides:
\(0.012 = 0.007x - 0.006x + 0.009\).
This simplifies to:
\(0.012 = 0.001x + 0.009\).

To isolate \(x\), subtract \(0.009\) from both sides:
\(0.012 - 0.009 = 0.001x\),
which simplifies to:
\(0.003 = 0.001x\).

Now we have isolated \(x\), making it easier to find its value.

checking solutions

After finding a solution for \(x\), it's crucial to check if it satisfies the original equation. This step ensures that no mistakes were made during the calculations.

We found that \(x = 3\). Substitute \(x = 3\) back into the original equation:
\(0.006(3 + 2) = 0.007(3) + 0.009\).

Simplify both sides:
\(0.006(5) = 0.021 + 0.009\)
which simplifies further to:
\(0.03 = 0.03\).

Since both sides are equal, the solution \(x = 3\) is correct. Always remember to check your solutions to confirm their accuracy.

step-by-step solving

Breaking the problem into smaller, manageable steps is key to solving linear equations. Here’s a quick review of the steps we took:

• Step 1: Use the distributive property to simplify the equation.
• Step 2: Move terms involving the variable to one side of the equation.
• Step 3: Isolate the variable.
• Step 4: Solve for the variable.
• Step 5: Check the solution to make sure it satisfies the original equation.

Each step builds on the previous one, making it easier to solve complex equations. Always take your time with each step to ensure accuracy.