Problem 33 - Prealgebra

Question

Graph the points \((-3,2)\) and \((1,3),\) and draw a line through them. Use that graph to answer Problems 31-34. Does the ordered pair (4, 5) lie on this line? (GRAPH CANT COPY)

Step-by-Step Solution

Answer

No, the ordered pair (4, 5) does not lie on the line.

Step 1: Identify Given Points

We are given two points to graph: - Point A at \((-3, 2)\)- Point B at \((1, 3)\). We'll start by marking these points on a coordinate plane.

Step 2: Plot the Points

On the graph, locate the point where the x-coordinate is -3 and the y-coordinate is 2 to plot Point A. Similarly, locate the point where the x-coordinate is 1 and the y-coordinate is 3 to plot Point B. These are the points through which our line will pass.

Step 3: Draw the Line

Using a ruler or straightedge, draw a straight line that passes through both Points A and B. This line represents all of the points that satisfy the linear equation derived from these two points.

Step 4: Confirm the Line's Equation

Determine the equation of the line by calculating the slope, \(m\), using the formula for slope: \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - 2}{1 - (-3)} = \frac{1}{4} \]Using Point B \((1, 3)\), the equation can be written in point-slope form, \(y - 3 = \frac{1}{4}(x - 1)\). Converting to slope-intercept form gives: \[ y = \frac{1}{4}x + \frac{11}{4} \].

Step 5: Check the Ordered Pair

Substitute the coordinates of the point (4, 5) into the line's equation to see if it satisfies the equation.\[ 5 = \frac{1}{4}(4) + \frac{11}{4} \]Calculate the right side: \[ \frac{4}{4} + \frac{11}{4} = \frac{15}{4} \]Comparing, 5 is not equal to \(\frac{15}{4}\). Thus, the point (4, 5) does not lie on the line.

Key Concepts

Coordinate Plane

The coordinate plane is a two-dimensional plane that helps us plot points and visualize relationships between them. It's made up of two perpendicular lines: the x-axis and the y-axis. These axes intersect at the origin, located at the point (0,0). The x-axis runs horizontally, while the y-axis runs vertically. This plane is divided into four quadrants. Each quadrant represents a unique combination of positive and negative values for x and y:

• Quadrant I: Both coordinates are positive.
• Quadrant II: x is negative, y is positive.
• Quadrant III: Both coordinates are negative.
• Quadrant IV: x is positive, y is negative.

Understanding how the coordinate plane is structured helps you accurately place points and sketch lines representing equations.

Plotting Points

Plotting points is the first step in graphing linear equations. To plot a point on the coordinate plane, you use an ordered pair (x, y). The x-value tells you how far to move horizontally from the origin. The y-value indicates how far to move vertically.
For example, for the point (-3, 2), move 3 units left from the origin and 2 units up. Mark this point. For the point (1, 3), move 1 unit right from the origin and 3 units up.

• Start from the origin (0,0).
• Move according to your x and y values.
• Mark the location with a dot.
• Label the points for clarity.

This skill is crucial because it's the foundation for both graphing equations and interpreting graphical data.

Slope of a Line

The slope of a line describes its steepness and direction. It's calculated using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1}\), where \(\ (x_1, y_1) \) and \((x_2, y_2)\) are any two distinct points on the line. Simply put, the slope is the change in y divided by the change in x.
The slope can be interpreted as:

• Positive: Line rises from left to right.
• Negative: Line falls from left to right.
• Zero: Line is horizontal.
• Undefined: Line is vertical.

In our example, using points (-3, 2) and (1, 3), the slope is \(\frac{1}{4}\). This tells us the line rises 1 unit for every 4 units it moves horizontally. Knowing how to calculate and interpret slope is vital when working with different equations and analyzing graphs.

Slope-Intercept Form

The slope-intercept form is a way of writing linear equations so they are easy to graph. The equation form is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. The y-intercept is where the line crosses the y-axis.
For the equation derived in our exercise, \(y = \frac{1}{4}x + \frac{11}{4}\), \(\frac{1}{4}\) is the slope and \(\frac{11}{4}\) is the y-intercept. With this format, you can quickly identify and graph the line by:

• Starting at the y-intercept on the y-axis.
• Using the slope to determine the direction and steepness of the line.

By expressing equations in slope-intercept form, you make it straightforward to visualize the line. It's a helpful tool for easily communicating and understanding the relationship described by the equation.