Question

In Exercises \(27-30,\) find the (a) slope and (b) \(y\) -intercept, and (c) graph the line. $$3 x+4 y=12$$

Short answer

The slope of the line is \(-\frac{3}{4}\), the y-intercept is \(3\) and the graph of the line starts at \(3\) on the y-axis and descends from left to right with a slope of \(-\frac{3}{4}\).

Step-by-step solution

Convert to Slope-Intercept Form

You start by converting the equation \(3 x+4 y=12\) into slope-intercept form. To do this, you need to solve for \(y\). First, you subtract \(3x\) from both sides, resulting in \(4y = -3x + 12\). Then, you divide each term by \(4\) to isolate \(y\), resulting in \(y = -\frac{3}{4}x + 3\).

Identify the Slope

The slope-intercept form of the equation is \(y = mx + c\) where \(m\) is the slope. In our equation, which is now \(y = -\frac{3}{4}x + 3\), the coefficient of \(x\) is \(-\frac{3}{4}\). Hence, the slope is \(-\frac{3}{4}\).

Identify the Y-Intercept

Returning to the slope-intercept format of the equation, where \(c\) is the point where the graph crosses the \(y\)-axis (also known as the y-intercept), in our equation \(y = -\frac{3}{4}x + 3\), the value of \(c\) is \(3\). Therefore, the y-intercept is \(3\).

Plot the Line on the Graph

To draw the line, first, you plot the y-intercept on the y-axis, which is the point (0,3). After that, use the slope to determine the direction and steepness of the line. The slope \(-\frac{3}{4}\) means that for every 4 units moving right along the x-axis, you move 3 units down on the y-axis. Repeat the process to plot a few more points and then draw a line through these points.

Key concepts

Slope of a Line

Understanding the slope of a line is a fundamental skill in algebra. The slope represents the rate of change and the direction of a line on a graph. It is denoted as 'm' in the slope-intercept form of a linear equation, which is written as \( y = mx + b \). The slope is calculated as the rise (the change in y) over the run (the change in x) between two points on the line. For example, with the equation \( y = -\frac{3}{4}x + 3 \), the slope is \( -\frac{3}{4} \). This means that for each unit you move to the right along the x-axis, the line falls 0.75 units (3/4 of a unit) vertically. A positive slope means the line ascends from left to right, while a negative slope means it descends. Identifying the slope allows you to predict and understand the behavior of the line on a graph, making it easier to visualize and represent relationships in various fields, such as economics and physics.

If an equation is not already in slope-intercept form, like \( 3x + 4y = 12 \), you would need to rearrange it as illustrated in the original exercise to identify the slope.

Y-Intercept

The y-intercept of a line indicates where the line crosses the y-axis of a coordinate plane. It is represented by the variable 'b' in the slope-intercept equation \( y = mx + b \). The y-intercept is a single point with coordinates \((0, b)\). This means that when the value of x is 0, y equals the y-intercept. In our example, the equation in slope-intercept form is \( y = -\frac{3}{4}x + 3 \), hence the y-intercept is 3, corresponding to the point \((0, 3)\) on the graph. Identifying the y-intercept is an essential step in graphing a line because it serves as a starting point. From this fixed point, you use the slope to determine other points on the line. The y-intercept is not only crucial for drawing the graph but also can represent an initial value or starting condition in real-world scenarios, like the starting amount in a bank account before interest is applied.

When working with linear equations, always remember to find the point where the line crosses the y-axis, as this will significantly simplify the graphing process.

Graphing Linear Equations

Graphing linear equations is a visual way to represent solutions to linear equations. The graph of any linear equation will always be a straight line. The process begins by first putting the equation into the slope-intercept form, \( y = mx + b \). Using the example \( y = -\frac{3}{4}x + 3 \), you then plot the y-intercept on the y-axis, which in this case is the point \((0, 3)\).

From there, you apply the slope. Since the slope is \( -\frac{3}{4} \), you move from the y-intercept 4 units to the right along the x-axis and 3 units down for each subsequent point. Connect these points with a straight line, and you have graphed the equation. Graphing helps in picturing the slope and intercepts, allows for visualization of algebraic concepts, and is a valuable tool in predicting and analyzing trends, making it a critical competency for students to master in mathematics and science subjects.

For students to fully grasp graphing linear equations, practice with a variety of slopes and y-intercepts is recommended, alongside drawing graphs for better visualization and understanding.