Problem 67 - Introductory and Intermediate Algebra for College Students 4th

Question

Write an equation in slope-intercept form of the line satisfying the given conditions. What is the slope of a line that is parallel to the line whose equation is \(A x+B y=C, B \neq 0 ?\)

Step-by-Step Solution

Answer

The slope of a line that is parallel to the line whose equation is \(Ax + By = C\) is \(-A/B\).

Step 1: Transform the given line to slope-intercept form

The first step is to convert the line \(Ax + By = C\) into the slope-intercept form \(y = mx + b\). To do this, isolate 'y': subtract 'Ax' from both sides of the equation, and then divide the whole equation by 'B'. This gives us \(y = -A/Bx + C/B\).

Step 2: Determine the slope of the line

In the equation obtained in step 1, the coefficient of 'x' is the slope of the line. In this case, the slope is \(-A/B\).

Step 3: Apply the parallel line rule

The slope of the line parallel to the given line is the same as the slope of the given line. Therefore, the slope of the line parallel to \(Ax + By = C\) is also \(-A/B\).

Key Concepts

Equation of a Line

Understanding the equation of a line is crucial for analyzing linear relations in algebra. The slope-intercept form is one of the most common ways to express a line's equation and is written as \(y = mx + b\), where \(m\) represents the slope of the line, and \(b\) denotes the y-intercept. The y-intercept is the point where the line crosses the y-axis (when \(x = 0\)).

To convert any linear equation to the slope-intercept form, we need to solve for \(y\). This often involves algebraic manipulation such as moving terms from one side of the equation to the other and dividing by coefficients. Students should be comfortable with operations like addition, subtraction, multiplication, division, and distributing terms across parentheses. These skills are necessary for isolating \(y\) and revealing the slope and y-intercept.

Slope of a Parallel Line

When addressing the slope of parallel lines, recall that parallel lines have identical slopes. That's a golden rule in geometry that links parallelism with consistency in steepness. If you're given the equation of a line and asked to find the slope of another line parallel to it, the task simplifies to extracting the slope from the first line's equation.

For instance, if the first line's equation is in slope-intercept form—\(y = mx + b\)—the parallel line has the same \(m\). When it's not, like with the standard form \(Ax + By = C\), we convert it to slope-intercept form, where the slope is the negative coefficient of \(x\) divided by the coefficient of \(y\), written as \(-A/B\). This conversion allows us to see the slope directly and verify parallel lines.

Algebraic Manipulation

Algebraic manipulation is the art of reshaping equations to reveal valuable insights like the slope or intercepts of a line. This process involves working with the properties of equality, distributive property, and rules for dealing with fractions. Basic steps include adding or subtracting terms to both sides of an equation, dividing or multiplying both sides by a coefficient, and factoring expressions when necessary.

When converting an equation like \(Ax + By = C\) into slope-intercept form, subtract \(Ax\) on both sides to get \(By = -Ax + C\). Next, divide everything by \(B\) to isolate \(y\), resulting in \(y = -A/Bx + C/B\). These algebraic steps are foundational for comprehending and manipulating equations in various forms and are widely applied in solving geometric and algebraic problems.