Problem 50 - College Algebra with Corequisite Support

Question

Starting with the point-slope formula \(y-y_{1}=m\left(x-x_{1}\right),\) solve this expression for \(x\) in terms of \(x_{1}, y, y_{1},\) and \(m\).

Step-by-Step Solution

Answer

\(x = \frac{y-y_{1} + mx_{1}}{m}\)

Step 1: Understand the Goal

We start with the point-slope formula for a straight line, given by \(y-y_{1}=m(x-x_{1})\). Our goal is to solve this equation for \(x\) in terms of \(x_{1}, y, y_{1},\) and \(m\). This means we need to isolate \(x\) on one side of the equation.

Step 2: Expand the Formula

First, expand the right side of the equation. Distribute \(m\) into \(x-x_{1}\), which changes the equation to \(y-y_{1}=mx-mx_{1}\).

Step 3: Move Terms Involving x

We want to isolate the term with \(x\), so move other terms to the opposite side of the equation. Add \(mx_{1}\) to both sides to get \(y-y_{1} + mx_{1} = mx\).

Step 4: Isolate x

Now, to solve for \(x\), divide every term by \(m\): \(x = \frac{y-y_{1} + mx_{1}}{m}\). This gives us the x-value in terms of \(x_{1}, y, y_{1},\) and \(m\).

Key Concepts

Solving for x

To solve for \( x \) in an equation, we need to reorganize the terms so that \( x \) stands alone on one side. This process often involves using inverse operations, such as addition to counter subtraction, or division to counter multiplication.
In the context of the point-slope formula \( y-y_{1}=m(x-x_{1}) \), solving for \( x \) requires a few steps:

• First, expand the expression by distributing the \( m \) over both \( x \) and \( x_1 \). This gives \( mx - mx_1 \).
• Next, bring all terms not involving \( x \) to the other side. For our equation, add \( mx_1 \) to both sides to balance the equation.
• Finally, divide all terms by \( m \) to isolate \( x \). This means dividing the entire equation by \( m \), thus solving the equation for \( x \).

Breaking down each step makes it easier to perform algebraic manipulations and achieve the desired variable isolation.

Linear Equations

Linear equations are polynomials of the first degree—this means the highest power of the variable(s) is 1. These equations take the form \( y = mx + b \), representing a straight line when graphed on a coordinate plane. The point-slope formula \( y-y_1 = m(x - x_1) \) is derived from the linear equation format and highlights the slope \( m \) and a specific point \( (x_1, y_1) \) on the line.
The slope \( m \) indicates the steepness and direction of the line:

• A positive slope means the line rises from left to right.
• A negative slope means it falls from left to right.

Understanding linear equations is fundamental because they model relationships in everyday problems, like calculating speed (distance over time) or predicting expenses (cost times quantity). By manipulating these equations, we can find the value of unknown variables, like \( x \), through algebraic processes.

Algebraic Manipulation

Algebraic manipulation involves using mathematical operations—addition, subtraction, multiplication, and division—to rearrange and simplify equations. The goal is often to isolate a variable, as in solving for \( x \). Key techniques include:

• Combining like terms: Group and simplify terms that have the same variables.
• Distributive property: Multiply a single term by each term inside a grouping, such as converting \( m(x - x_1) \) to \( mx - mx_1 \).
• Inverse operations: Used to reverse operations around a variable, like adding to counter subtraction.

By practicing these techniques, you can simplify and solve complex equations step-by-step. In our example, applying these steps systematically gets \( x \) on one side, allowing us to express it in terms of given variables. Algebraic manipulation is a powerful tool for unlocking solutions to an array of mathematical challenges.