Problem 74 - Introductory Algebra for College Students

Question

graph each linear equation in two variables. Find at least five solutions in your table of values for each equation. $$y=-\frac{3}{2} x+2$$

Step-by-Step Solution

Answer

The graph of the equation \(y=-\frac{3}{2}x+2\) is a downward sloping line crossing the \(y\)-axis at point (0,2). The five solutions {(-2,5), (-1,3.5), (0,2), (1,0.5), (2,-1)} lie on this line.

Step 1: Understand the Linear Equation

The given equation is \(y=-\frac{3}{2} x+2\). We can interpret it as: the value of \(y\) decreases by 3/2 units for each increase in 1 unit of \(x\). The \(+2\) represents the \(y\)-intercept, which means the line crosses the \(y\)-axis at point (0,2).

Step 2: Construct a Table of Values

Start by choosing arbitrary values for \(x\). Simply substitute each value of \(x\) into the equation to find the corresponding \(y\) value. Here is a proposed selection (x,y): (-2,5), (-1,3.5), (0,2), (1,0.5), (2,-1).

Step 3: Plot the Points on the Graph

Plot each pair of points from the table of values on a graph. Ensure the points align correctly with the \(x\) and \(y\) axis.

Step 4: Draw the Line

Once all points are plotted, connect them using a straight line. This is the graphical representation of the given equation.

Key Concepts

Linear Equation Solutions

Linear equations are foundational in algebra and can be visualized through graphing on a coordinate plane. A linear equation typically presents a relationship between two variables, most commonly 'x' and 'y'. Solving a linear equation involves finding the set of all possible pairs of 'x' and 'y' that make the equation true.
These solution pairs can be represented as points on a graph, and when connected, they form a straight line, hence the term 'linear'. The linear equation in the exercise, \(y = -\frac{3}{2}x + 2\), indicates that for every unit increase in 'x', the value of 'y' decreases by 1.5 units (\(\frac{3}{2}\)), revealing a negative slope. The constant term '+2' tells us where the line crosses the y-axis, known as the y-intercept.

Table of Values

A table of values is a pragmatic method of turning an algebraic expression into something visual and often more comprehensible. To create a table of values for a linear equation like \(y = -\frac{3}{2}x + 2\), you select inputs for 'x' and calculate the corresponding outputs for 'y'.
Let's take a closer look:

• Select values for 'x'—these can be any numbers, but typically we choose numbers around the origin (0,0) for ease of plotting.
• Substitute each 'x' value into the equation to solve for 'y'.
• Record these 'x' and 'y' pairs in a two-column table. Each pair is a solution to the equation.

Creating such a table not only aids in visualizing solutions but also prepares you for graphing by providing concrete points to plot.

Slope-Intercept Form

The slope-intercept form is a straightforward way to write linear equations and readily provides important graphical information. It is written as \(y = mx + b\), where 'm' represents the slope and 'b' indicates the y-intercept. In our example, \(y = -\frac{3}{2}x + 2\), the slope (m) is \(-\frac{3}{2}\) and the y-intercept (b) is 2.
Understanding the slope:

• The slope 'm' tells us how steep the line is and in which direction it tilts. A negative slope means the line inclines downwards from left to right.
• The numerator of the slope fraction shows how many units 'y' changes, and the denominator shows the change in 'x'. Here, for every 2 units of change in 'x', 'y' decreases by 3 units.

Appreciating the y-intercept:

• The y-intercept 'b' is the point where the line crosses the y-axis. It is the value of 'y' when 'x' is zero.
• In graphs, it allows us to begin plotting from a known location, conveniently kickstarting the graphing process.

The slope-intercept form is particularly user-friendly because it gives a clear instruction on plotting the line: Start at the y-intercept and follow the slope.