Reviewing algebra concepts

Full Answer Section

            Step 3: Calculate the change in $x$ (x2​−x1​). Change in $x=7-3=4$ Step 4: Divide the change in $y$ by the change in $x$. m=46​ Step 5: Simplify the fraction. m=23​ Therefore, the slope of the line that passes through points A(3, -2) and B(7, 4) is 23​. 3. Describe What the Slope Value Represents: The slope value of m=23​ in this case represents the following:

• Positive Slope: Since the value is positive, it indicates that the line is moving upwards from left to right. As you move along the line from the left to the right, the $y$-value increases as the $x$-value increases.
• Steepness: The numerical value of 23​ (or 1.5) indicates the steepness of the line. For every 2 units you move horizontally to the right (the "run"), the line moves 3 units vertically upwards (the "rise"). A larger absolute value of the slope would mean a steeper line, while a smaller absolute value would mean a flatter line.
• Direction: The line is generally moving in an upward, rightward direction on a coordinate plane.

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Part B: Analyzing the New Slope with Point B'

  1. What is the new slope of the line between A(3, -2) and B'(7, -2)? Given the new points: A (x1​,y1​)=(3,−2) B' (x2​,y2​)=(7,−2) Step 1: Identify the coordinates. x1​=3 y1​=−2 x2​=7 y2​=−2 Step 2: Calculate the change in $y$ (y2​−y1​). Change in $y=-2-(-2)=-2+2=0$ Step 3: Calculate the change in $x$ (x2​−x1​). Change in $x=7-3=4$ Step 4: Divide the change in $y$ by the change in $x$. m=40​ Step 5: Simplify the fraction. $m=0$ The new slope of the line between A(3, -2) and B'(7, -2) is $0$. 2. How does this new slope compare to the one in Part A? The new slope ($m=0$) is significantly different from the slope in Part A (m=23​).

• In Part A, the slope was positive and had a numerical value of 1.5, indicating an upward-sloping line.
• In Part B, the slope is exactly zero, indicating a line with no vertical change.

3. What kind of line is formed in this case? Please explain why. In this case, a horizontal line is formed. Explanation: A horizontal line is formed because the $y$-coordinates of both points A and B' are the same (both are -2). This means there is no change in the vertical position (the "rise") between the two points. When the change in $y$ is zero, the slope will always be zero, provided there is a change in $x$. A slope of zero is the defining characteristic of a horizontal line.    

Sample Answer

       

Part A: Finding the Slope Between A(3, -2) and B(7, 4)

  1. Explain the Formula for Calculating Slope: The slope ($m$) of a line that passes through two points (x1​,y1​) and (x2​,y2​) is a measure of its steepness and direction. It represents the "rise" (change in $y$-coordinates) over the "run" (change in $x$-coordinates). The formula for calculating slope is: m=Change in xChange in y​=x2​−x1​y2​−y1​​ 2. Find the Slope and Show Each Step in Detail: Given the two points: A (x1​,y1​)=(3,−2) B (x2​,y2​)=(7,4) Now, substitute these values into the slope formula: Step 1: Identify the coordinates. x1​=3 y1​=−2 x2​=7 y2​=4 Step 2: Calculate the change in $y$ (y2​−y1​). Change in $y=4-(-2)=4+2=6$