Problem 27 - Elementary Algebra

Question

For Problems 1-36, graph each linear equation. (Objective 2) $$ 4 x+5 y=-10 $$

Step-by-Step Solution

Answer

Graph the line by plotting y-intercept \(-2\) and using the slope \(-\frac{4}{5}\) to find another point, then draw the line through these points.

Step 1: Write the Equation in Slope-Intercept Form

The given equation is \(4x + 5y = -10\). To graph the equation, we will first convert it into slope-intercept form, \(y = mx + b\), where \(m\) represents the slope and \(b\) the y-intercept. Solve for \(y\):\[5y = -4x - 10\]\[y = -\frac{4}{5}x - 2\]

Step 2: Identify the Slope and Y-Intercept

From the equation \(y = -\frac{4}{5}x - 2\), we identify the slope \(m\) as \(-\frac{4}{5}\) and the y-intercept \(b\) as \(-2\). This tells us that the line crosses the y-axis at \(y = -2\).

Step 3: Plot the Y-Intercept on the Graph

Begin by plotting the y-intercept of \(-2\) on the y-axis of the graph. This is your starting point for graphing the line.

Step 4: Use the Slope to Determine another Point

The slope \(-\frac{4}{5}\) means that for every 5 units moved to the right (positive x-direction), the line moves 4 units down (negative y-direction). From the point \((0, -2)\), move 5 units to the right to \((5, -2)\), and then 4 units down to \((5, -6)\). Plot the point \((5, -6)\) on the graph.

Step 5: Draw the Line

Connect the y-intercept point \((0, -2)\) and the point \((5, -6)\) with a straight line. This line represents the graph of the equation \(4x + 5y = -10\). Extend the line in both directions, adding arrows to indicate it continues infinitely.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a very common way to express linear equations. It is given by the formula \(y = mx + b\). Here, \(m\) stands for the slope of the line, while \(b\) is the y-intercept where the line crosses the y-axis. This form makes it easy to quickly identify the key characteristics of the line, such as how steep it is and where it starts on the y-axis.
To convert an equation into the slope-intercept form, you need to solve for \(y\). For example, starting with the equation \(4x + 5y = -10\), isolating \(y\) involves a few simple algebraic steps: rearrange terms to get \(5y = -4x - 10\), and then divide each term by 5 to find \(y = -\frac{4}{5}x - 2\). This newly formed equation directly shows you both the slope and the y-intercept.

Y-Intercept

The y-intercept in the equation of a line is the point where the line crosses the y-axis. In the slope-intercept form \(y = mx + b\), this is represented by the \(b\) value. This point is crucial because it provides a starting location for plotting the line on a graph. In our example, the equation \(y = -\frac{4}{5}x - 2\) shows that the y-intercept is \(-2\).
To find the y-intercept from a given equation, you look for the constant term \(b\) in the equation when it is in slope-intercept form. This point is easy to plot as it will always have the coordinates \( (0, b)\), substituting zero for the \(x\)-value to remain on the y-axis.

Plotting Points

Once the y-intercept is marked, the next step in graphing a linear equation involves plotting points that the line will pass through. Using the slope helps you efficiently determine where the other points lie.
The slope \(m\) in the equation \(y = -\frac{4}{5}x - 2\) tells us how the line moves: a slope of \(-\frac{4}{5}\) indicates that for every 5 units you move horizontally to the right, you move 4 units down vertically. Starting from the y-intercept \((0, -2)\), you go right 5 units to \((5, -2)\) and down 4 units to arrive at the point \((5, -6)\).
Plotting these points and connecting them gives you a clear representation of the linear equation on the graph. This visual method not only confirms your calculations but also provides a comprehensive understanding of the line's direction and steepness.