Problem 58 - Algebra for College Students

Question

Solve each equation, and check the solution. \(-\frac{7}{8} x=6\)

Step-by-Step Solution

Answer

x = -\frac{48}{7}

Step 1: Isolate the Variable

To isolate the variable on one side of the equation, multiply both sides of the equation by the reciprocal of \(-\frac{7}{8}\). This will cancel out the fraction on the left side.

Step 2: Multiply by the Reciprocal

The reciprocal of \(-\frac{7}{8}\) is \(-\frac{8}{7}\). Multiply both sides by \(-\frac{8}{7}\). \[\left( -\frac{8}{7} \right) \left( -\frac{7}{8} x \right) = \left( -\frac{8}{7} \right) (6)\]\[ \frac{56}{56} x = -\frac{48}{7} \]\[x = -\frac{48}{7} \]

Step 3: Simplify the Solution

Simplify \(-\frac{48}{7} \) if needed. In this case, it is already in its simplest form. So, \ x = -\frac{48}{7} \.

Step 4: Check the Solution

Substitute \ x = -\frac{48}{7} \ back into the original equation to verify. \[ -\frac{7}{8} \left( -\frac{48}{7} \right) = 6 \]Simplifying the left side yields:\[ \frac{7 \cdot 48}{8 \cdot 7} = 6 \]\[ \frac{48}{8} = 6 \]\[ 6 = 6 \]The original equation holds true, so the solution is verified.

Key Concepts

Isolating the Variable

When solving linear equations, the first priority is often to isolate the variable you are solving for. This means getting the variable by itself on one side of the equation. To do this, we'll need to perform operations that eliminate other numbers or terms from that side. For example, in the equation \(-\frac{7}{8} x=6\), the variable is \(x\) and our goal is to get \(x\) by itself. A common strategy is to multiply both sides of the equation by the reciprocal of the coefficient attached to the variable. The reciprocal of a fraction is simply flipping its numerator and denominator. Isolating the variable helps make further steps in solving the equation simpler and more straightforward.

Multiplying by the Reciprocal

In our example, the coefficient of \(x\) is \(-\frac{7}{8}\). To cancel out \(-\frac{7}{8}\), we multiply both sides of the equation by its reciprocal, \(-\frac{8}{7}\). This step is crucial because it allows us to eliminate the fraction, resulting in only the variable \(x\)Remaining on one side of the equation:
We perform the following operation:
\[ \left( -\frac{8}{7} \right) \left( -\frac{7}{8} x \right) = \left( -\frac{8}{7} \right) (6) \]
With this, we get:
\[ x = -\frac{48}{7} \]
Notice that multiplying by the reciprocal changed the left side into one (since any number times its reciprocal is one), effectively isolating \(x\). This clearly shows how multiplying by the reciprocal transforms a challenging equation into a simpler one.

Simplifying Fractions

After isolating the variable, you may need to simplify the resulting fraction. Simplifying means reducing the fraction to its simplest form where the numerator and denominator have no common factors other than 1. In our example, \(x = -\frac{48}{7}\) is already in its simplest form because 48 and 7 have no common divisible other than 1. However, if you needed to simplify a more complex fraction, you would divide both the numerator and the denominator by their greatest common divisor (GCD). Simplifying fractions can make your final answer cleaner and easier to understand.

Verifying Solutions

The final step in solving any equation is to verify that your solution is correct. This is done by substituting your solution back into the original equation and checking that both sides of the equation are equal. For our example, we substitute \ x = -\frac{48}{7} \:
\[ -\frac{7}{8} \left( -\frac{48}{7} \right) = 6 \]
Simplifying the left side, we get:
\[ \frac{7 \cdot 48}{8 \cdot 7} = 6 \]
Which simplifies further to:
\[ \frac{48}{8} = 6 \]
\[ 6 = 6 \]
Since both sides of the equation are equal, the solution is verified to be correct. Verification ensures that you haven't made any mistakes during your calculations and instills confidence that your solution is accurate.