Problem 89 - Algebra A Combined Function

Question

Solve. $$ 0.035 x+5.112=0.010 x+5.107 $$

Step-by-Step Solution

Answer

The solution is \( x = -0.2 \).

Step 1: Isolate the Variable Terms

To isolate the variable terms, we need to move the term with the variable on the right side of the equation to the left. Subtract \(0.010x\) from both sides of the equation:\[ 0.035x + 5.112 - 0.010x = 5.107 \]This simplifies to:\[ 0.025x + 5.112 = 5.107 \]

Step 2: Isolate the Constant Terms

Next, we need to move all constant terms to the right side of the equation. Subtract \(5.112\) from both sides:\[ 0.025x + 5.112 - 5.112 = 5.107 - 5.112 \]This simplifies to:\[ 0.025x = -0.005 \]

Step 3: Solve for the Variable

To solve for \(x\), divide both sides of the equation by \(0.025\):\[ x = \frac{-0.005}{0.025} \]Simplify the fraction:\[ x = -0.2 \]

Step 4: Solution Verification

Substitute \(x = -0.2\) back into the original equation to verify:Original equation:\[ 0.035(-0.2) + 5.112 = 0.010(-0.2) + 5.107 \]Calculate each side:- Left: \(0.035(-0.2) = -0.007 \), so \(-0.007 + 5.112 = 5.105 \)- Right: \(0.010(-0.2) = -0.002 \), so \(-0.002 + 5.107 = 5.105 \)Both sides are equal. Thus, the solution is verified.

Key Concepts

Linear Equations

Linear equations are mathematical expressions that create a straight line when graphed. They are usually written in the form \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants and \(x\) is the variable. These equations are fundamental in algebra as they help in determining the value of the unknown variable that makes the equation true. They play a vital role in various fields such as physics, engineering, and economics for modeling relationships and predicting outcomes. Understanding linear equations provides a strong foundation for solving more complex mathematical problems. When dealing with linear equations, it's important to remember that the solutions are valid only if the mathematical operations are applied consistently to both sides of the equation. This ensures the balance of equality.

Solving Equations

Solving equations involves finding the value of the variable that satisfies the equation. In algebra, solving linear equations typically follows a structured approach:

• Identify the equation and simplify both sides if necessary.
• Apply mathematical operations to isolate the variable term on one side.
• Perform the reverse of the operations used, maintaining equation balance.
• Check the solution by substituting the variable back into the original equation.

By following these steps, you ensure that each transformation correctly maintains the equality, helping you arrive at the correct value for the variable. It's crucial to pay attention to details like signs and arithmetic operations to avoid errors.

Variable Isolation

Variable isolation is a method used to rearrange the equation so that the unknown variable stands alone on one side of the equation. This process often requires:

• Subtracting or adding terms to both sides to move the variable.
• Dividing or multiplying values to further simplify the expression.

In our original exercise, we started by subtracting \(0.010x\) from both sides to group the variable terms together. This operation results in the simplified form allowing us to manipulate the equation until \(x\) was isolated. Consistent application of inverse operations ensures that the variable truly becomes isolated, making it possible to determine its value.

Equation Verification

Equation verification is the final step in the problem-solving process where you confirm that the obtained solution is indeed correct. This involves substituting the solution back into the original equation.For example, in the exercise solution, after determining that \(x = -0.2\), you plug this value into the original equation to check balance on both sides. Calculate each side separately and compare results:

• Left Side: Calculate with the solution substituted into the left-hand expression.
• Right Side: Calculate with the solution substituted into the right-hand expression.

If both sides of the equation are equal, the solution is verified as correct. This step is crucial because it not only confirms the accuracy but also ensures understanding of the solution process.