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How to Find the Slope-Intercept Equation of a Line from a Graph
Mathematics
Grade 8
Question Content
Write the equation of this line in slope-intercept form. Write your answer using integers, proper fractions, and improper fractions in simplest form.
Correct Answer
y = \\frac{1}{3}x - 3
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, -3), so b = -3.
3
Step 3: Calculate the slope (m) using two clear points on the line. Use the y-intercept (0, -3) and another point like (6, -1). The slope formula is m = \\frac{y_2 - y_1}{x_2 - x_1}. Substitute the points: m = \\frac{-1 - (-3)}{6 - 0} = \\frac{2}{6} = \\frac{1}{3}.
4
Step 4: Substitute m and b into the slope-intercept formula: y = \\frac{1}{3}x - 3.
Knowledge Points Involved
1
Slope-Intercept Form of a Line
The slope-intercept form of a linear equation is y = mx + b, where m represents the slope (rate of change of y with respect to x) and b represents the y-intercept (the y-coordinate of the point where the line intersects the y-axis, at x=0). This form is used to quickly graph a line and write its equation when slope and y-intercept are known.
2
Slope Calculation
The slope m of a line passing through two points (x₁, y₁) and (x₂, y₂) is calculated using the formula m = \\frac{y_2 - y_1}{x_2 - x_1}. It measures how steep the line is; a positive slope means the line rises from left to right, while a negative slope means it falls.
3
Y-Intercept Identification
The y-intercept is the point where a line crosses the y-axis, which always has an x-coordinate of 0. It can be directly read from a graph by finding the y-value at x=0, and it is represented by the constant term b in the slope-intercept equation.
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