Problem 65 - Introductory and Intermediate Algebra for College Students 4th

Question

Graph each linear equation in two variables. Find at least five solutions in your table of values for each equation. $$y=-3 x-1$$

Step-by-Step Solution

Answer

The five solutions for the equation \(y=-3x-1\) are (-2, 5), (-1, 2), (0, -1), (1, -4), and (2, -7). These can be plotted to graph the given linear equation, resulting in a descending straight line due to the negative slope.

Step 1: Determine Five Solutions

To find five solutions, we can start by arbitrarily choosing five x-values and then plugging them into the equation to find their corresponding y-values. For example, let's take -2, -1, 0, 1, 2 as the x-values and substitute them into \(y=-3x-1\) to find the y-values.

Step 2: Calculate Y-Values

Substitute the chosen x-values into the equation. When x=-2, \(y=-3\times(-2)-1=5\) When x=-1, \(y=-3\times(-1)-1=2\) When x=0, \(y=-3\times 0 -1=-1\) When x=1, \(y=-3\times 1 -1=-4\) When x=2, \(y=-3\times 2 -1=-7\) So, the pairs are (-2, 5), (-1, 2), (0, -1), (1, -4), and (2, -7).

Step 3: Graph the Equation

Plot the coordinate pairs on the plane and draw a line passing through all the points. Remember that a linear equation in two variables graphs a straight line. The line for this problem should descend ('negative slope'), and the points should fall along this line.

Key Concepts

Linear Equations

Linear equations are mathematical expressions that model relationships between two variables, typically written in the form \( y = mx + b \). Here, \( m \) represents the slope of the line while \( b \) is the y-intercept, the point where the graph crosses the y-axis. This equation is called 'linear' because its graph is always a straight line.
When graphing a linear equation like \( y = -3x - 1 \), you can identify the behavior of the line in terms of its slope and intercept. For example, the equation shows that the slope \( m \) is -3, indicating a downward or negative trend as the line moves from left to right. The intercept \( b \) is -1, showing where the line meets the y-axis.
Understanding linear equations is crucial because they form the backbone of many algebraic concepts. They appear in contexts ranging from simple physics problems to complex data modeling. Recognizing their form and function will help make interpreting and creating such relations more intuitive.

Coordinate Plane

The coordinate plane, also known as the Cartesian plane, is essential for graphing equations. It consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). Where these axes intersect is called the origin, denoted by (0, 0).
This plane allows us to plot points defined by pairs of numbers. Each point in this plane corresponds to an ordered pair \( (x, y) \). For example, the point (2, -7) is plotted by moving 2 units to the right of the origin (since x is positive) and 7 units down (since y is negative). Understanding how to navigate and use the coordinate plane is key to graphing equations like \( y = -3x - 1 \).

• Understanding the coordinate plane helps visualize algebraic relationships.
• It forms a foundation for more advanced graphing, including graphs of non-linear relationships.

Using the coordinate plane effectively allows one to road map the relationships dictated by algebraic expressions, turning abstract concepts into visual and tangible solutions.

Slope

Slope is one of the critical elements defining the behavior of a line in a linear equation. It quantifies the steepness and direction of a line on the coordinate plane. The slope \( m \) is calculated as the 'rise' over the 'run', or the change in the y-coordinates divided by the change in the x-coordinates of two distinct points on a line. In mathematical terms, if you have two points \((x_1, y_1)\) and \((x_2, y_2)\), the slope \( m \) can be given by the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \).
A positive slope indicates a line rising from left to right, while a negative slope shows a line descending. Zero slope means the line is horizontal, and an undefined slope suggests a vertical line. In the equation \( y = -3x - 1 \), the slope is -3, signifying that for every 1 unit increase in x, y decreases by 3 units.
Understanding slopes gives insight into how changes in one variable affect the other. It's not just about the numbers, but the relationship they articulate visually on the graph. With this concept, you can predict and interpret trends both in mathematics and real-world scenarios.