Problem 4 Use the given equation to comple... [FREE SOLUTION] (2024)

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When working with linear equations, you often come across ordered pairs. An ordered pair is a pair of numbers written in a specific order, typically as (x, y). The first number represents the x-coordinate and the second number represents the y-coordinate. In this exercise:

• We had ordered pairs like (0, __), (4, __), and (-4, __).

To find the missing y-values, you substitute the x-values into the given equation. For example, to find the y-value for x = 0 in the equation \(y = -\frac{3}{4} x + 2\):

• Replace x with 0: \(y = -\frac{3}{4}(0) + 2 = 2\).

This gives us the ordered pair (0, 2). Similarly, you can find the y-values for x = 4 and x = -4, resulting in the ordered pairs (4, -1) and (-4, 5), respectively.

To graph the ordered pairs, you need to understand the coordinate plane. The coordinate plane is a two-dimensional surface where each point is determined by an ordered pair (x, y). It has two perpendicular lines called axes:

• The horizontal line is the x-axis.
• The vertical line is the y-axis.

Where these lines intersect is the origin, with coordinates (0, 0). Points on the coordinate plane are plotted as follows:

• Start at the origin (0, 0).
• Move along the x-axis to the x-coordinate.
• Then, move parallel to the y-axis to the y-coordinate.

For instance, to plot the point (4, -1):

• Move 4 units to the right (positive direction) along the x-axis.
• Then move 1 unit down (negative direction) along the y-axis.

Repeat the process for each ordered pair to create the graph.

Linear functions are mathematical functions that form a straight line when graphed. These functions can be expressed in the form y = mx + b, where:

• m is the slope of the line.
• b is the y-intercept, the point where the line crosses the y-axis.

The equation \(y = -\frac{3}{4} x + 2\) is an example of a linear function. Here:

• The slope (m) is -\frac{3}{4}. This tells us that for every increase of 4 units in x, y decreases by 3 units.
• The y-intercept (b) is 2, meaning the line crosses the y-axis at (0, 2).

Linear functions have consistent rates of increase or decrease across their domain, making them predictable and easy to graph once you have the slope and the y-intercept.

The slope-intercept form is a specific way of writing the equation of a line. It's written as: \(y = mx + b\). This form is highly useful because it directly provides two key characteristics of the line: the slope and the y-intercept.

• Slope (m): This is a measure of how steep the line is. In the equation \(y = -\frac{3}{4} x + 2\), the slope is -\frac{3}{4}, indicating the line falls 3 units for every 4 units it moves to the right.
• Y-Intercept (b): This is where the line crosses the y-axis. In this exercise, the y-intercept is 2, meaning the line crosses at the point (0, 2).

To graph a line using slope-intercept form:

• Plot the y-intercept first.
• From the y-intercept, follow the slope to find another point on the line.
• Draw a straight line through these points to extend it in both directions.

Understanding this form makes graphing linear equations straightforward and intuitive. By just knowing the slope and y-intercept, you can quickly sketch the corresponding line on a coordinate plane.