Problem 75 - Algebra 2

Question

Solve each equation. $$ 6=\frac{s-8}{-7} $$

Step-by-Step Solution

Answer

The solution is \(s = -34\).

Step 1: Eliminate the Fraction

First, multiply both sides of the equation by -7 to eliminate the fraction. This gives us the equation: \(-7 \times 6 = s - 8\), which simplifies to \(-42 = s - 8\).

Step 2: Solve for 's'

Now, we need to isolate \(s\) on one side of the equation. To do so, add 8 to both sides of the equation: \(-42 + 8 = s\).

Step 3: Simplify

Simplify the expression \(-42 + 8\): This results in \(-34 = s\).

Step 4: Final Solution

Thus, the solution to the equation \(6 = \frac{s-8}{-7}\) is \(s = -34\).

Key Concepts

Fractions

Fractions are numbers that represent a part of a whole and are written in the form of a numerator over a denominator. In mathematics, fractions can often appear as part of equations.
When solving equations with fractions, it's essential to eliminate these fractions to simplify the process. This is typically done by finding a common denominator or by multiplying both sides of the equation by the denominator of the fraction you want to eliminate.

• For example, in the equation \(6 = \frac{s-8}{-7}\), -7 is the denominator.
• Multiplying both sides by -7 helps us remove the fraction, transforming the equation to a simpler form: \(-7 \times 6 = s - 8\).

This step is crucial as it results in a linear equation that is easier to solve.

Linear Equations

Linear equations are algebraic expressions that involve constants and variables raised only to the first power. These equations are usually represented in the form \(ax + b = c\) where \(a\), \(b\), and \(c\) are constants and \(x\) is a variable.
The objective of solving a linear equation is to find the value of the variable that makes the equation true. They are called "linear" because when graphed, they create a straight line.

• In our example, after removing the fraction, we have the linear equation: \(-42 = s - 8\).
• Solving involves manipulating terms to get the variable 's' alone on one side.

Linear equations are among the most straightforward equations to solve, making them an important foundation in algebra.

Isolating Variables

Isolating variables is a critical step in solving linear equations. It involves rearranging the equation so the variable you are solving for is by itself on one side.
This can be done by using inverse operations such as addition, subtraction, multiplication, and division.

• For instance, with the equation \(-42 = s - 8\), our goal is to solve for \(s\).
• We add 8 to both sides to isolate \(s\): \(-42 + 8 = s\).
• This simplifies to \(-34 = s\), providing the solution.

Isolating the variable helps in seeing explicitly what the solution should be, making it a powerful technique in solving equations efficiently.