Problem 34 - Introductory Algebra for College Students

Question

Solve each equation using the addition property of equality. Be sure to check your proposed solutions. $$r+\frac{3}{5}=-\frac{7}{10}$$

Step-by-Step Solution

Answer

The solution to the equation \(r+\frac{3}{5}=-\frac{7}{10}\) is \(r = -\frac{13}{10}\)

Step 1: Isolate the variable 'r'

Subtract \(\frac{3}{5}\) from both sides of the equation. This will isolate the variable 'r'. This is done using the addition property of equality which states that we can add or subtract the same value from both sides of an equation and the equality will still hold. The operation gives \( r = -\frac{7}{10} - \frac{3}{5}\)

Step 2: Calculate the value of 'r'

Calculate the right side of the expression by subtracting the fractions. \( -\frac{7}{10} - \frac{3}{5} = -\frac{7}{10} - \frac{6}{10} = -\frac{13}{10}\)

Step 3: Check the solution

Substitute \(r = -\frac{13}{10}\) into the original equation to verify the solution. \[ r + \frac{3}{5} = -\frac{13}{10} + \frac{6}{10} = -\frac{7}{10}\] Since the left side equals the right side, the solution is correct.

Key Concepts

Solving Linear Equations

When we talk about solving linear equations, we mean finding the value of the variable that makes the equation true. The process usually involves a series of steps that can include adding, subtracting, multiplying, or dividing both sides of the equation by the same number.
In our example, the linear equation is in the form of a variable, in this case, 'r', plus a fraction equals a negative fraction. After applying operations equally on both sides, we can solve for the unknown variable. Solving linear equations is a fundamental skill in algebra, and understanding the principles behind it can help you tackle more complex mathematical problems.

Isolating Variables

Isolating variables is a crucial step in solving linear equations. The goal is to get the variable on one side of the equation by itself. To do this, we perform operations that 'undo' what is being done to the variable.
For instance, if a variable is being added by a number or another variable, we subtract that amount from both sides. Conversely, if a variable is being subtracted, we'd add. If the variable is being multiplied, we divide both sides, and if it's divided, multiply. Throughout these steps, we maintain the balance of the equation by doing the same thing to both sides, a principle known as the addition (or subtraction) property of equality, which allows us to isolate the variable and solves for it effectively.

Subtracting Fractions

Common Denominators and Subtraction

The major step in subtracting fractions is finding a common denominator for the fractions involved. This means adjusting the fractions so that they both have the same bottom number. Once you have common denominators, you can easily subtract the numerators (the top numbers), and the common denominator remains the same.
In the exercise given, we encountered fractions with denominators of 5 and 10. By converting the fractions to have a common denominator, which is 10 in this case, we are able to subtract the numerators easily. Remember, when working with negative signs, it's also important to pay attention to their proper placement to get the correct result.

Equation Verification

Once we think we've solved an equation, it's vital to verify our answer to ensure it is correct—this is what we call equation verification. To do this, we substitute our solution back into the original equation and check if both sides remain equal.
If they do, our solution is verified; if not, we may have made a mistake and need to review our steps. Equation verification is an essential part of problem-solving because it not only confirms the accuracy of our work but also reinforces our understanding of the underlying mathematical principles.