Problem 36 - College Algebra with Corequisite Support

Question

For the following exercises, given each set of information, find a linear equation satisfying the conditions, if possible. \(x\) intercept at (-5,0) and \(y\) intercept at (0,4)

Step-by-Step Solution

Answer

The equation is \( y = \frac{4}{5}x + 4 \).

Step 1: Understand Intercepts

The intercepts given are the points at which the line crosses the axes. The x-intercept is (-5,0), meaning when y=0, x=-5. The y-intercept is (0,4), meaning when x=0, y=4.

Step 2: Use Intercepts to Identify Slope

The slope (m) of a line can be found using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Using the intercepts, the slope is \( m = \frac{4 - 0}{0 - (-5)} = \frac{4}{5} \).

Step 3: Write Equation in Slope-Intercept Form

The slope-intercept form of a line is given by \( y = mx + b \), where m is the slope and b is the y-intercept. Plug in the slope \( \frac{4}{5} \) and y-intercept 4: \( y = \frac{4}{5}x + 4 \).

Step 4: Confirm Equation Fits Conditions

Check that the equation \( y = \frac{4}{5}x + 4 \) satisfies both intercepts. - For x = -5, y = 0: \( 0 = \frac{4}{5}(-5) + 4 = -4 + 4 = 0 \). Satisfied.- For y = 4, x = 0: \( 4 = \frac{4}{5}(0) + 4 = 4 \). Satisfied.

Key Concepts

Understanding the X-Intercept

The x-intercept of a linear equation is the point where the graph of the equation crosses the x-axis. At this point, the value of y is always zero. In simple terms, it represents the value of x when y equals zero. For example, in our exercise, the x-intercept is given as (-5,0). This means that when y = 0, the corresponding value of x is -5. Understanding this concept is crucial, as it provides one of the points needed to describe the line on a graph. Knowing the x-intercept helps in sketching the graph and establishing the characteristics of the linear equation.

Exploring the Y-Intercept

The y-intercept of a linear equation is where the graph intersects the y-axis. At this point, the value of x is zero. Essentially, it tells us the y-value when x equals zero. For the linear equation in our problem, the y-intercept is (0,4). This indicates that w hen x is 0, the y-value is 4, meaning the line touches the y-axis at point 4. Recognizing the y-intercept is important as it often serves as the starting point when graphing a line, and is directly used when writing the equation of the line in slope-intercept form.

The Slope-Intercept Form Explained

The slope-intercept form of a linear equation is written as \( y = mx + b \), where \( m \) represents the slope of the line, and \( b \) is the y-intercept. This form is very straightforward and efficient for graphing. In this format, you can easily see both where the line will cross the y-axis (the y-intercept) and how steep the line will be (the slope). Our example equation from the exercise becomes \( y = \frac{4}{5}x + 4 \), where 4 is the y-intercept and \( \frac{4}{5} \) is the slope. This form helps quickly identify the line's characteristics without plotting points one by one.

Calculating the Slope

The slope of a line indicates how steep the line is and the direction in which it moves. You can calculate the slope using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). It is essentially the 'rise over run' between two points on a line. In this exercise, the two points (-5,0) and (0,4) are the x-intercept and y-intercept, respectively. Plugging these points into the formula gives \( m = \frac{4 - 0}{0 - (-5)} = \frac{4}{5} \). Thus, the slope of the line is \( \frac{4}{5} \), signifying it rises 4 units for every 5 units it runs. Understanding slope calculations is vital for determining how to draw the line accurately and construct its equation.