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How to Find the Slope, Equation, and Graph of a Line Through Points A(1,2) and B(5,6)
Mathematics
Grade 9 of Junior High School
Question Content
Given two points A(1,2) and B(5,6): 1. Calculate the slope of the line passing through the points. 2. Find the equation of the line in slope-intercept form (y=mx+c). 3. Plot points A and B, then draw the line through them.
Correct Answer
1. Slope m=1; 2. Line equation: y = x + 1; 3. Graph with points A(1,2) and B(5,6) connected by a straight line
Detailed Solution Steps
1
Step 1: Calculate the slope (m) using the slope formula for two points (x₁,y₁) and (x₂,y₂): m = (y₂ - y₁)/(x₂ - x₁). Substitute A(1,2) as (x₁,y₁) and B(5,6) as (x₂,y₂): m = (6 - 2)/(5 - 1) = 4/4 = 1.
2
Step 2: Use the slope-intercept form y = mx + c, substitute m=1 and the coordinates of point A(1,2) into the equation to solve for c: 2 = 1(1) + c. Simplify to get 2 = 1 + c, then subtract 1 from both sides: c = 2 - 1 = 1. Substitute m=1 and c=1 back into the slope-intercept form to get the line equation: y = x + 1.
3
Step 3: On a coordinate plane, locate point A at (1,2) (1 unit right on the x-axis, 2 units up on the y-axis) and point B at (5,6) (5 units right on the x-axis, 6 units up on the y-axis). Use a straight edge to draw a line connecting these two points, extending it beyond both points to represent the full line.
Knowledge Points Involved
1
Slope Formula for Two Points
The formula m=(y₂-y₁)/(x₂-x₁) calculates the steepness (slope) of a straight line passing through two points (x₁,y₁) and (x₂,y₂). It is used to quantify how much the y-value changes per unit change in the x-value, and is the first step in finding the equation of a line from two points.
2
Slope-Intercept Form of a Line
The slope-intercept form is written as y = mx + c, where m is the slope of the line, and c is the y-intercept (the point where the line crosses the y-axis, at x=0). This form is useful for quickly graphing a line and understanding its key properties, and is derived by substituting a known point and slope to solve for the y-intercept c.
3
Graphing a Line from Two Points
To graph a line from two given points, first plot each point on a coordinate plane by matching the x-coordinate to the horizontal axis and y-coordinate to the vertical axis. Then, draw a straight line through the two points, extending it in both directions. This works because only one unique straight line can pass through two distinct points.
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