Problem 82 - Introductory and Intermediate Algebra for College Students 4th

Question

Explain how to graph \(A x+B y=C\) if \(C\) is not equal to zero.

Step-by-Step Solution

Answer

The graph of \(Ax + By = C\) can be created by first isolating y, which results in \(y = -\frac{A}{B}x + \frac{C}{B}\). Then identify the slope and y-intercept, which are \(-\frac{A}{B}\) and \(\frac{C}{B}\) respectively, then plot the y-intercept on the graph and use the slope to find another point. Connecting the two points gives the graph of the equation.

Step 1: Write the equation

We start with the equation \(Ax + By = C\).

Step 2: Isolate y in terms of x

To isolate y, first subtract \(Ax\) from both sides, that result in \(By = -Ax + C\). Then, divide every term by \(B\), so the equation becomes \(y = -\frac{A}{B}x + \frac{C}{B}\)

Step 3: Identify the slope and y-intercept

The equation is now in the form y = mx + b. The coefficient of x is the slope of the line, and the constant term is the y-intercept. So, the slope, \(m\), is \( -\frac{A}{B}\) and the y-intercept, \(b\), is \(\frac{C}{B}\).

Step 4: Plot the graph

After identifying the slope (m) and the y-intercept (b), plot the y-intercept on the y-axis. Then, use the slope to find another point on the line. The slope tells us to move vertically -A unit(s) and horizontally B unit(s) from the y-intercept. Once we've plotted a second point, we can draw a straight line connecting the two points, which represents the graph of the given equation.

Key Concepts

Slope-intercept form

The slope-intercept form is a way of expressing the equation of a straight line. It is written as \( y = mx + b \). Here, \( m \) represents the slope of the line, and \( b \) is the y-intercept. This form is incredibly useful because it immediately reveals both the slope and the y-intercept, making graphing on a coordinate plane straightforward. Graphs can be quickly sketched by starting at the y-intercept and using the slope to find other points.

Coordinate plane

A coordinate plane is a two-dimensional surface where we can graph lines and curves. It is made up of a horizontal axis, known as the x-axis, and a vertical axis, called the y-axis. These axes meet at a point called the origin. On this plane, each point has a pair of coordinates, \( (x, y) \), which specify its location relative to the axes. The coordinate plane allows us to visually represent equations like lines intersecting or parallel lines based on their equations.

Slope and y-intercept

In the equation \( y = mx + b \), the slope \( m \) defines the steepness and direction of the line. A positive slope means the line rises from left to right, while a negative slope means it falls. The value of \( m \) is calculated as the change in \( y \) divided by the change in \( x \) between any two points on the line. The y-intercept \( b \) is where the line crosses the y-axis, and it tells us the value of \( y \) when \( x = 0 \). Knowing both slope and y-intercept is essential for accurately graphing linear equations on the coordinate plane.

Linear function graphing

Graphing a linear function involves plotting its equation on the coordinate plane. Start with the y-intercept \( b \), which is the point where the line meets the y-axis. From this point, use the slope \( m \) to find other points. If \( m = \frac{3}{2} \), for example, move 3 units up for each 2 units right from the y-intercept to mark the next point. Connect these points with a straight line, extending infinitely in both directions. Graphing helps to visualize the relationship between variables represented by the linear equation.