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How to Write a Line's Equation in Fully Simplified Slope-Intercept Form from a Graph
Mathematics
Grade 8 (Junior High School)
Question Content
Write the equation of the line in fully simplified slope-intercept form, using the provided graph of a line on a coordinate plane.
Correct Answer
y = x - 2
Detailed Solution Steps
1
Step 1: Recall that slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept (the point where the line crosses the y-axis).
2
Step 2: Identify the y-intercept (b) from the graph: the line crosses the y-axis at (0, -2), so b = -2.
3
Step 3: Calculate the slope (m) using two clear points on the line, such as (0, -2) and (2, 0). Use the slope formula m = (y₂ - y₁)/(x₂ - x₁). Substitute the points: m = (0 - (-2))/(2 - 0) = 2/2 = 1.
4
Step 4: Substitute m = 1 and b = -2 into the slope-intercept formula to get the equation: y = 1x - 2, which simplifies to y = x - 2.
Knowledge Points Involved
1
Slope-Intercept Form of a Line
The slope-intercept form is written as y = mx + b, where m represents the slope (rate of change of the line) and b represents the y-intercept (the y-coordinate of the point where the line intersects the y-axis). This form is used to quickly write the equation of a line when the slope and y-intercept are known, and it makes it easy to graph a line by starting at the y-intercept and using the slope to find additional points.
2
Slope Calculation
The slope (m) of a line between two points (x₁, y₁) and (x₂, y₂) is calculated using the formula m = (y₂ - y₁)/(x₂ - x₁). The slope measures how steep the line is: a positive slope means the line rises from left to right, a negative slope means it falls, a slope of 0 means it is horizontal, and an undefined slope means it is vertical.
3
Y-Intercept Identification
The y-intercept is the point where a line crosses the y-axis of a coordinate plane. This point always has an x-coordinate of 0, so it is written as (0, b) where b is the y-intercept value. It can be directly read from a graph by finding where the line intersects the vertical y-axis.
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