Problem 54 - Understanding Elementary Algebra with Geometry

Question

Sketch the graph of the given equation. Label the intercepts. $$\frac{y+6}{3}=\frac{x-4}{4}$$

Step-by-Step Solution

Answer

The y-intercept is at (0, -\frac{34}{3}) and the x-intercept is at (8.5, 0).

Step 1: Rewrite the Equation in Slope-Intercept Form

Start by rewriting the given equation \[\frac{y + 6}{3} = \frac{x - 4}{4}\]This can be rewritten as \[3(y + 6) = 4(x - 4)\] Next, distribute the constants: \[3y + 18 = 4x - 16\] Now, solve for y: \[3y = 4x - 16 - 18\]\[3y = 4x - 34\] Divide by 3: \[y = \frac{4}{3}x - \frac{34}{3}\]Hence, the slope-intercept form is \[y = \frac{4}{3}x - \frac{34}{3}\]

Step 2: Identify the y-Intercept

The y-intercept can be found by setting x = 0 in the slope-intercept form: \[y = \frac{4}{3}(0) - \frac{34}{3}\] \[y = -\frac{34}{3}\] Thus, the y-intercept is at \left(0, -\frac{34}{3} \right)\.

Step 3: Identify the x-Intercept

The x-intercept is found by setting y = 0 in the slope-intercept form: \[0 = \frac{4}{3}x - \frac{34}{3}\] Isolate for x by adding \(\frac{34}{3}\): \[\frac{34}{3} = \frac{4}{3}x\] Multiply both sides by 3: \[34 = 4x\] Solve for x: \[x = \frac{34}{4}\] \[x = 8.5\] Thus, the x-intercept is at (8.5, 0).

Step 4: Graph the Line

Using the slope \(\frac{4}{3}\) and the intercepts found, plot the line on a coordinate plane. Start at the y-intercept \(0, -\frac{34}{3}\)\ and use the slope (rise over run) to plot another point or directly plot the x-intercept at (8.5, 0). Draw a straight line through these points.

Step 5: Label the Intercepts

Make sure to label the x-intercept at (8.5, 0) and the y-intercept at \(0, -\frac{34}{3}\)

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is one of the most common ways to represent a line. It is written as: \[ y = mx + b \] where:

• m is the slope of the line, which shows how steep the line is.
• A positive slope means the line goes upwards as you move from left to right.
• A negative slope means the line goes downwards.
• b is the y-intercept, the point where the line crosses the y-axis.

To put an equation into slope-intercept form, you need to solve it for \(y\). This method makes it easier to graph because you can immediately identify the slope and the y-intercept.

y-Intercept

The y-intercept is where the line crosses the y-axis. To find it:

• Set \(x = 0\) in the equation.
• Solve for \(y\).

For the equation, \(\frac{y+6}{3} = \frac{x-4}{4}\), after converting it to slope-intercept form, it becomes:
\[ y = \frac{4}{3}x - \frac{34}{3} \] Setting \(x = 0\), we get:
\[ y = \frac{4}{3}(0) - \frac{34}{3}\] \[ y = -\frac{34}{3} \] Thus, the y-intercept is at \( (0, -\frac{34}{3}) \). This means the line crosses the y-axis at \(y = -\frac{34}{3}\).

x-Intercept

The x-intercept is where the line crosses the x-axis. To find it:

• Set \(y = 0\) in the equation.
• Solve for \(x\).

Using the slope-intercept form of the equation:
\[ 0 = \frac{4}{3}x - \frac{34}{3} \] We add \(\frac{34}{3}\) to both sides:
\[ \frac{34}{3} = \frac{4}{3}x \] Multiplying both sides by 3: \[ 34 = 4x \] Solve for \(x\): \[ x = \frac{34}{4} \] \[ x = 8.5 \] Thus, the x-intercept is at \((8.5, 0)\). This means the line crosses the x-axis at \(x = 8.5\).

Coordinate Plane

The coordinate plane, or Cartesian plane, is a two-dimensional surface on which we can plot points, lines, and curves. It is defined by two perpendicular axes:

• The x-axis (horizontal)
• The y-axis (vertical)

Each point on the plane is specified by a pair of coordinates \((x, y)\), where \(x\) is the position on the horizontal axis and \(y\) is the position on the vertical axis. When graphing linear equations:

• Start by plotting the intercepts.
• Use the slope \(m\) to find additional points.

In our example, start by plotting the y-intercept \((0, -\frac{34}{3})\) and the x-intercept \((8.5, 0)\). Draw a straight line through these points to represent the equation.