Problem 32 - Introductory Algebra for College Students

Question

Solve each equation and check your proposed solution in Exercises. Begin your work by rewriting each equation without fractions. $$\frac{x}{2}+13=-22$$

Step-by-Step Solution

Answer

The solution to the equation is \(x=-70\).

Step 1: Removing the Fraction

Start by multiplying every term by 2 to get rid of the fraction in the equation. Doing this, the equation becomes \(2*(x/2)+2*13=2*-22\), which simplifies to \(x+26=-44\).

Step 2: Solve for the Variable

Next, isolate x by subtracting 26 from both sides of the equation. This gives the equation: \(x=-44-26\).

Step 3: Simplify Equation

Simplify the right side of the equation to find the solution to x, which is -70.

Step 4: Check the Solution

Substitute x=-70 back into the original equation to confirm the solution. The left side of the equation will become \( \(-70/2)+13\), which is -22. As this is equal to the right side of the original equation, the solution is verified to be correct.

Key Concepts

Understanding Fractional Equations

Fractional equations are algebraic expressions that include fractions. To solve these, one common method is to eliminate the fractions as the first step. This is often done by finding the least common denominator (LCD) of all the fractions involved and multiplying each term of the equation by this LCD.

For example, in the equation \(\frac{x}{2}+13=-22\), the only fraction has a denominator of 2. Thus, by multiplying every term by 2, we easily get rid of the fractional part converting it into a simpler linear equation. This helps to avoid mistakes with fractions as we proceed to find the value of the variable.

How to Isolate the Variable in an Equation

Once you've eliminated fractions, the next step in solving an equation is to isolate the variable. This means moving all terms with the variable to one side of the equation and all the constant terms to the other side.

In the equation \(x + 26 = -44\), we want to isolate \(x\) by getting rid of the constant term attached to it. This is done by performing the inverse operation. Since 26 is added to \(x\), we subtract 26 from both sides resulting in \(x = -44 - 26\).

Isolating the variable is a critical step which makes it easier to see the solution clearly. It is essentially undoing everything that is being done to the variable until it stands alone.

Verifying Algebraic Solutions

The final and often overlooked step in solving algebraic equations is to verify the solution. This step is critical to ensure that the solution is correct.

To verify a solution, simply substitute the value of the variable back into the original equation. In our example, we substitute \(x = -70\) back into the original equation \(\frac{x}{2} + 13 = -22\). If the left-hand side simplifies to -22, which matches the right-hand side, we have confirmed that -70 is indeed the solution.

Verification not only confirms the correctness of our solution but also helps students to understand the importance of checking their work. It's crucial to form this as a regular habit in problem-solving.