Problem 47 - Algebra 2

Question

Graph each equation. $$ 4 x-3 y=-6 $$

Step-by-Step Solution

Answer

The line passes through the points (0, 2) and (3, 6).

Step 1: Convert equation to slope-intercept form

Rearrange the given equation to solve for y in terms of x. The slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept.

Step 2: Solve for y-intercept

Isolate y on one side of the equation to find the y-intercept (the value of y when x=0).

Step 3: Solve for another point using a different x-value

Choose a different value for x and solve for y to find another point on the graph.

Step 4: Plot the points and draw the line

Plot the y-intercept and the second point found on the Cartesian plane, then draw a straight line through them, extending the line across the plane.

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form is essential for graphing linear equations efficiently. The slope-intercept form is an equation of a straight line expressed as \( y = mx + b \).

In this equation, \( m \) represents the slope of the line which indicates the steepness and direction of the line. It can be thought of as the 'rise over run', or the change in \( y \) (vertical) for each unit of change in \( x \) (horizontal). Meanwhile, \( b \) indicates the y-intercept which is the point where the line crosses the y-axis.

When graphing an equation such as \( 4x - 3y = -6 \), the first step is to rearrange it into the slope-intercept form. By moving terms around we get \( y = \frac{4}{3}x + 2 \), where \( \frac{4}{3} \) is the slope (\( m \) value) and \( 2 \) is the y-intercept (\( b \) value). This form instantly provides the necessary information to graph the line quickly and accurately.

Y-Intercept

The y-intercept is a crucial point on the Cartesian plane that represents where a line crosses the y-axis. In the slope-intercept equation \( y = mx + b \), the y-intercept is given by the value of \( b \).

For the equation we're considering, \( 4x - 3y = -6 \), after rearranging it into the slope-intercept form, we found the y-intercept to be \( 2 \). This means that when \( x \) is equal to \( 0 \), the value of \( y \) will be \( 2 \). To plot this on a graph, you simply locate the point (0, 2) on the y-axis.

The y-intercept is a starting point for graphing the entire line. Once it's plotted, you can use the slope to determine another point on the line and draw the complete line through these two points.

Cartesian Plane

The Cartesian plane, also known as the coordinate plane, consists of two perpendicular number lines that intersect at the origin (0,0). These lines are known as the x-axis (horizontal) and y-axis (vertical).

Every point on the plane can be described by a pair of numerical coordinates: \( (x, y) \), which indicate the point's location relative to the origin. When graphing the line \( y = \frac{4}{3}x + 2 \), you would start by plotting the y-intercept at the point (0, 2). Next, using the slope, or the 'rise over run', you would rise 4 units in the y direction and run 3 units in the x direction from the y-intercept, to find another point on the line.

Connecting these two points with a straight line and extending it across the plane creates the graph of the linear equation on the Cartesian plane. This visual representation is a powerful tool for understanding linear relationships and solving algebraic problems involving linear equations.