Problem 54 - College Algebra Essentials

Question

Find all values of \(x\) satisfying the given conditions. \(y_{1}=\frac{x+1}{4}, y_{2}=\frac{x-2}{3},\) and \(y_{1}-y_{2}=-4\)

Step-by-Step Solution

Answer

The value of \(x\) that satisfies the given conditions is \(x = -37\).

Step 1: Substitute \(y_1\) and \(y_2\) using the given equations

Substitute \(y_1 = \frac{x+1}{4}\) and \(y_2 = \frac{x-2}{3}\) into the equation \(y_{1}-y_{2}=-4\). The equation becomes: \[ \frac{x+1}{4} - \frac{x-2}{3} = -4 \].

Step 2: Solve for \(x\)

First, put the equation over the common denominator (12 in this case) to simplify the equation: \[ \frac{3(x+1)-4(x-2)}{12} = -4 \]. Simplify the numerator: \[ \frac{3x+3 - 4x +8}{12} = -4 \]. Combine like terms in the numerator: \[ \frac{-x+11}{12} = -4 \]. And finally, solve for \(x\), which yields: \[ x = -48 + 11 = -37 \].

Key Concepts

Solving for x

Solving for \(x\) involves finding the value of the variable \(x\) that satisfies the given equation. This is often one of the main goals when dealing with linear equations.
Linear equations, like the one we worked with here, are equations where the variable is raised to the power of one. They are the simplest type of equations, and solving them generally requires the isolation of the variable.
The process is straightforward:

• First, substitute known values, as exhibited by using \(y_1 = \frac{x+1}{4}\) and \(y_2 = \frac{x-2}{3}\).
• Next, rearrange the equation to isolate \(x\), using algebraic techniques.
• Perform any calculations needed to simplify the equation.

This systematic approach helps to determine the specific value of \(x\) that makes the original equation true, which in our case was \(x = -37\).

Algebraic Manipulation

Algebraic manipulation is fundamental in math and refers to the steps we take to simplify or solve equations.
In our equation, algebraic manipulation is shown through several important activities:

• Substitution: We substituted \(y_1\) and \(y_2\) into the given equation.
• Simplifying expressions: The expression \( \frac{3(x+1)-4(x-2)}{12} = -4 \) was simplified by combining like terms, resulting in \( \frac{-x+11}{12} = -4 \).
• Using arithmetic operations: We utilized addition, subtraction, and multiplication to work through the equation.

By carrying out these actions, we systematically reduced complexity, making \(x\) easier to solve. The key is to maintain balance in the equation by performing the same operations on both sides.

Fractions in Equations

When dealing with linear equations that include fractions, additional steps are necessary to simplify the equation. Fractions often make equations appear more complicated, but they can be managed just as readily with some extra care:

• Find a common denominator: For \( \frac{x+1}{4} - \frac{x-2}{3} = -4 \), we used 12 as a common denominator to simplify combining terms.
• Perform fractional arithmetic: Fractions can be combined by adjusting expressions over a common denominator, allowing for the removal of fractions by simplifying the numerators.
• Clear fractions for easier solving: This involves multiplying through by the common denominator to eliminate fractions entirely, if possible.

These techniques enable us to simplify equations involving fractions, ensuring the equation is easier to work with and solve. This process led us to find that \(x = -37\), resolving the initial equation.