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Find the Equation of a Line in y=mx+c Form from a Graph
Mathematics
Grade 9 (Junior High School)
Question Content
Write down the equation of the line drawn below. Give your answer in the form $y = mx + c$.
Correct Answer
$y = x - 2$
Detailed Solution Steps
1
Step 1: Identify the y-intercept ($c$). The y-intercept is the point where the line crosses the y-axis. Looking at the graph, the line crosses the y-axis at $(0, -2)$, so $c = -2$.
2
Step 2: Calculate the slope ($m$). The slope is the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. Choose two clear points on the line, e.g., $(0, -2)$ and $(2, 0)$. The rise is $0 - (-2) = 2$, and the run is $2 - 0 = 2$. So $m = \\frac{2}{2} = 1$.
3
Step 3: Substitute $m$ and $c$ into the slope-intercept form $y = mx + c$. This gives 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 of a straight line is $y = mx + c$, where $m$ represents the slope (steepness) of the line, and $c$ represents the y-intercept (the y-coordinate of the point where the line crosses the y-axis). This form is used to quickly write the equation of a line when its slope and y-intercept are known, and it is widely applied in graphing linear relationships and solving linear problems.
2
Y-Intercept Identification
The y-intercept is the point where a line intersects the y-axis, which always has an x-coordinate of 0. It is the constant term $c$ in the slope-intercept equation $y = mx + c$, and it tells the value of $y$ when $x = 0$.
3
Slope Calculation
The slope ($m$) of a line measures how much $y$ changes for a unit change in $x$. It is calculated as $m = \\frac{y_2 - y_1}{x_2 - x_1}$ using two distinct points $(x_1, y_1)$ and $(x_2, y_2)$ on the line. A positive slope means the line rises from left to right, while a negative slope means it falls.
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