Problem 47 - Understanding Elementary Algebra with Geometry

Question

Sketch the graph of the given equation. Label the intercepts. $$y=-18 x+633.6$$

Step-by-Step Solution

Answer

The graph intercepts are at (0, 633.6) and (35.2, 0).

Step 1 - Identify the Slope and Y-Intercept

The given equation is in the slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Here, \(m = -18\) and \(b = 633.6\). The slope indicates that for every unit increase in \(x\), \(y\) decreases by 18 units.

Step 2 - Find the Y-Intercept

The y-intercept occurs when \(x = 0\). Substituting \(x = 0\) into the equation gives \(y = 633.6\). So, the y-intercept is \((0, 633.6)\).

Step 3 - Find the X-Intercept

The x-intercept occurs when \(y = 0\). Set up the equation as follows: \[ 0 = -18x + 633.6 \].\ To solve for \(x\), isolate \(x\) by adding \(18x\) to both sides and then divide by 18: \[ 18x = 633.6 \] \[ x = \frac{633.6}{18} \] \( x = 35.2 \). So, the x-intercept is \((35.2, 0)\).

Step 4 - Plot the Intercepts

On a coordinate plane, plot the y-intercept \((0, 633.6)\) and the x-intercept \((35.2, 0)\). These points will help in sketching the graph.

Step 5 - Draw the Line

Draw a straight line through the points \((0, 633.6)\) and \((35.2, 0)\). This line represents the graph of the equation \(y = -18x + 633.6\).

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is a common way of writing the equation of a line. It is expressed as \(y = mx + b\), where:

• \(m\) represents the slope of the line
• \(b\) is the y-intercept, which is the point where the line crosses the y-axis

In our exercise, the given equation is \(y = -18x + 633.6\). Here, the slope \(m\) is -18, and the y-intercept \(b\) is 633.6. The slope tells us how steep the line is and the direction it goes. A negative slope, like -18, means the line slopes downwards as we move from left to right. The y-intercept is where the line hits the y-axis, meaning when \(x = 0\). In simpler terms,

• The slope-intercept form gives us a straightforward way to graph a linear equation.
• It shows how one variable changes in relation to the other.

X-Intercept

The x-intercept is the point where a graph crosses the x-axis. At this point, the value of y is always zero. To find the x-intercept of the equation \(y = -18x + 633.6\), we set \(y = 0\) and solve for \(x\). Let's break it down step by step:

• Start with the equation: \(0 = -18x + 633.6\)
• Add \(18x\) to both sides to isolate \(x\): \(18x = 633.6\)
• Divide by 18 to solve for \(x\): \(x = \frac{633.6}{18} = 35.2\)

So, the x-intercept is at the point (35.2, 0). This tells us that the line crosses the x-axis at x = 35.2. To sketch the graph, this point is plotted on the coordinate plane and helps us accurately draw the line.

• Remember: The x-intercept is always where \(y = 0\).
• It is found by solving the equation for \(x\) when \(y = 0\).

Y-Intercept

The y-intercept is the point where the line crosses the y-axis. At this point, the value of x is always zero. Finding the y-intercept is quite straightforward when using the slope-intercept form of a linear equation \(y = mx + b\). The y-intercept is simply the value of \(b\). For the equation \(y = -18x + 633.6\), the y-intercept \(b\) is 633.6. This means that the line crosses the y-axis at the point (0, 633.6). In other words:

• The y-intercept helps us start plotting the graph of a linear equation.
• It's the initial value of \(y\) when \(x = 0\).

With this point plotted on the coordinate plane, it sets the initial position for our line. The slope then determines the angle and direction of the line from this y-intercept.

• The y-intercept is always where \(x = 0\).
• It is found directly from \(b\) in the equation \(y = mx + b\).