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Find the Equation of a Line Parallel to a Given Line Through a Point
Mathematics
Grade 9 (Junior High School)
Question Content
The graph below shows point P and line A. Line B passes through point P and is parallel to line A. Work out the equation of line B. Give your answer in the form $y = mx + c$
Correct Answer
$y=2x-7$
Detailed Solution Steps
1
Step 1: Calculate the slope of Line A. From the graph, Line A passes through (0,-8) and (4,0). Use the slope formula $m = \\frac{y_2-y_1}{x_2-x_1}$: $m = \\frac{0-(-8)}{4-0} = \\frac{8}{4}=2$.
2
Step 2: Identify the coordinates of point P. From the graph, point P is at (3, -1).
3
Step 3: Use the fact that parallel lines have equal slopes, so Line B also has a slope $m=2$. Substitute $m=2$, $x=3$, $y=-1$ into the slope-intercept form $y=mx+c$ to solve for $c$: $-1 = 2(3) + c$, which simplifies to $-1=6+c$. Solve for $c$: $c=-1-6=-7$.
4
Step 4: Substitute $m=2$ and $c=-7$ into $y=mx+c$ to get the equation of Line B: $y=2x-7$.
Knowledge Points Involved
1
Slope of a straight line
The slope $m$ of a line passing through two points $(x_1,y_1)$ and $(x_2,y_2)$ is calculated by $m=\\frac{y_2-y_1}{x_2-x_1}$. It represents the steepness and direction of the line; a positive slope means the line rises from left to right, while a negative slope means it falls.
2
Slope property of parallel lines
In a coordinate plane, two non-vertical parallel lines have identical slopes. This is because parallel lines have the same steepness and direction, so their rate of change of $y$ with respect to $x$ is equal.
3
Slope-intercept form of a straight line
The slope-intercept form of a line is $y=mx+c$, where $m$ is the slope of the line and $c$ is the y-intercept (the y-coordinate of the point where the line crosses the y-axis). This form is used to write the equation of a line when the slope and a point on the line, or the slope and y-intercept, are known.
4
Reading coordinates from a graph
To find the coordinates of a point on a grid graph, read the horizontal value (x-coordinate) first, then the vertical value (y-coordinate). For a line, identify two clear points where it intersects grid lines to calculate its slope.
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