Problem 70 - Algebra for College Students

Question

Solve each equation, and check the solution. \(0.05 x+0.08+0.06 x=0.07 x+0.68\)

Step-by-Step Solution

Answer

x = 15

Step 1: Combine like terms on the left side

Identify and combine the like terms on the left side of the equation: \[0.05x + 0.08 + 0.06x = 0.07x + 0.68\] Combining the like terms involving x: \[ (0.05x + 0.06x) + 0.08 = 0.07x + 0.68 \] This simplifies to: \[0.11x + 0.08 = 0.07x + 0.68\]

Step 2: Move all x terms to one side

Subtract \(0.07x\) from both sides of the equation to move all terms involving x to one side: \[0.11x - 0.07x + 0.08 = 0.07x - 0.07x + 0.68\] This simplifies to: \[0.04x + 0.08 = 0.68\]

Step 3: Move constant terms to the other side

Subtract 0.08 from both sides of the equation to move the constant term to the right side: \[0.04x + 0.08 - 0.08 = 0.68 - 0.08\] This simplifies to: \[0.04x = 0.60\]

Step 4: Solve for x

Divide both sides of the equation by 0.04 to isolate x: \[ \frac{0.04x}{0.04} = \frac{0.60}{0.04} \] This simplifies to: \[x = 15\]

Step 5: Check the solution

Substitute the value of x back into the original equation to verify the solution: \[0.05(15) + 0.08 + 0.06(15) = 0.07(15) + 0.68\] Calculate each term: \[0.75 + 0.08 + 0.90 = 1.05 + 0.68\]This simplifies to: \[1.73 = 1.73\] Since both sides are equal, the solution is correct.

Key Concepts

Combining Like Terms

In algebra, 'like terms' are terms that have the same variable raised to the same power. Combining like terms helps simplify equations and makes it easier to solve them. In the given exercise: \[0.05x + 0.08 + 0.06x = 0.07x + 0.68\], we look for terms with the same variable, which here are the terms involving 'x'. So, we combine \(0.05x\) and \(0.06x\) to get: \(0.05x + 0.06x = 0.11x\). Thus, our equation now looks like this: \[0.11x + 0.08 = 0.07x + 0.68\]
This step significantly simplifies the problem and sets up the equation for the following steps.

Isolating Variables

Once we combine like terms, the next core step is isolating the variable we are solving for. The goal is to get the variable 'x' by itself on one side of the equation. So, starting with: \[0.11x + 0.08 = 0.07x + 0.68\], we move all terms involving 'x' to one side by subtracting \(0.07x\) from both sides: \[0.11x - 0.07x + 0.08 = 0.07x - 0.07x + 0.68\] Simplifying, we get: \[0.04x + 0.08 = 0.68\] Next, we move the constant term to the opposite side of the equation. Subtract 0.08 from both sides: \[0.04x + 0.08 - 0.08 = 0.68 - 0.08\] This simplifies to: \[0.04x = 0.60\]

Algebraic Manipulation

Algebraic manipulation involves adjusting the equation using valid algebraic steps to find the solution. Once the variable term is isolated: \[0.04x = 0.60\], we need to solve for 'x'. To do this, divide both sides of the equation by the coefficient of 'x' which is 0.04: \[\frac{0.04x}{0.04} = \frac{0.60}{0.04}\], Simplifying, we get: \[x = 15\]
This technique ensures that we maintain the equality and correctly solve for the variable.

Checking Solutions

Verifying our solution is vital to ensure accuracy. Substitute \(x = 15\) back into the original equation: \[0.05(15) + 0.08 + 0.06(15) = 0.07(15) + 0.68\] Calculate each term: \[0.75 + 0.08 + 0.90 = 1.05 + 0.68\] Which simplifies to: \[1.73 = 1.73\]
Since both sides are equal, the solution \(x = 15\) is correct. This step ensures our work is accurate and the solution is validated before concluding the problem.