Problem 39 - Elementary Algebra

Question

For Problems \(33-44\), determine the slope and \(y\) intercept of the line represented by the given equation, and graph the line. $$ -y=-\frac{3}{4} x+4 $$

Step-by-Step Solution

Answer

Slope: \(\frac{3}{4}\), Y-intercept: \(-4\).

Step 1: Rewrite the Equation in Slope-Intercept Form

The given equation is \[ -y = -\frac{3}{4}x + 4 \]First, we need to multiply the entire equation by -1 to isolate y on one side, leading to:\[ y = \frac{3}{4}x - 4 \] This is the slope-intercept form \(y = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept.

Step 2: Identify the Slope and Y-Intercept

From the rewritten equation \( y = \frac{3}{4}x - 4 \), identify the slope \(m\) and the y-intercept \(b\):- Slope \(m = \frac{3}{4}\)- Y-intercept \(b = -4\)This means the line has a slope of \(\frac{3}{4}\) and crosses the y-axis at \(y = -4\).

Step 3: Plot the Y-Intercept

Start by plotting the y-intercept on the graph. Since \(b = -4\), place a point at (0, -4) on the coordinate plane.

Step 4: Use the Slope to Plot Another Point

The slope \(\frac{3}{4}\) means that for every 3 units up, the line goes 4 units to the right. From the y-intercept (0, -4), move up 3 units to (0, -1), then 4 units to the right to (4, -1). Plot this point.

Step 5: Draw the Line

With the two points plotted, (0, -4) and (4, -1), draw a straight line through these points. This is the graphical representation of the equation \( y = \frac{3}{4}x - 4 \).

Key Concepts

Linear Equations

Linear equations are fundamental in mathematics because they graphically represent straight lines. The general form of a linear equation is given by \(ax + by = c\), which includes two variables, typically \(x\) and \(y\). To express a linear equation in a more useful and graphically intuitive way, we use the slope-intercept form:

• Slope-Intercept Form: \(y = mx + b\)

In this form, \(y\) is the dependent variable, \(x\) is the independent variable, \(m\) represents the slope, and \(b\) is the \(y\)-intercept.
Transforming a linear equation into the slope-intercept form makes it easy to visualize and solve. By looking at the equation \(y = mx + b\), you can quickly identify the slope and \(y\)-intercept, which assists in graphing the line on a coordinate plane.
It's important to understand these concepts to grasp how linear relationships behave and how changes in \(m\) and \(b\) affect the line.

Slope

The slope of a linear equation is a number that measures its steepness or tilt. It's represented by \(m\) in the slope-intercept form \(y = mx + b\). In practical terms, the slope tells us how much \(y\) changes when \(x\) increases by one unit. Slopes can be expressed as a fraction, whole number, or even zero.

• Positive Slope: When \(m > 0\), the line rises as it moves from left to right.
• Negative Slope: When \(m < 0\), the line falls as it moves from left to right.
• Zero Slope: When \(m = 0\), the line is horizontal, showing constant \(y\) values regardless of \(x\).

Understanding slope is crucial for analyzing how a linear graph behaves. In the context of the exercise, the slope \(\frac{3}{4}\) indicates for every 4 horizontal units, the line rises 3 vertical units.
This is key in plotting additional points after identifying the \(y\)-intercept.

Y-Intercept

The \(y\)-intercept in a linear equation is the point where the line crosses the \(y\)-axis. It's represented by \(b\) in the slope-intercept form \(y = mx + b\). To find or verify the \(y\)-intercept on a graph:

• Fixed Point: The \(y\)-intercept is located at \( (0, b) \) on the graph, meaning \(x\) is always zero here.
• Visual Aid: It provides a starting point for plotting the line, as it’s the initial value of \(y\) when \(x = 0\).

In our exercise example, the \(y\)-intercept is \(-4\), which means the line crosses the \(y\)-axis at \( (0, -4) \).
This position is crucial as it serves as the starting point for graphing the equation. After plotting the \(y\)-intercept point, the slope information helps in determining additional points on the graph. Understanding the \(y\)-intercept helps students to quickly draw or interpret linear graphs.