// Numbas version: finer_feedback_settings {"name": "Straight lines: one point gradient", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Straight lines: one point gradient", "tags": ["graph", "linear", "linear equation", "Straight Line", "straight line", "y=mx+b"], "metadata": {"description": "

Given one point and the gradient determine the equation of the line.

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The gradient-intercept form of the line that passes through the point $(\\var{point_x},\\var{point_y})$ with a gradient of $\\simplify{{rise}/{run}}$ is $y=$ [[0]].

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There are two common ways to approach these questions:

Use the one-point gradient formula $y-y_1=m(x-x_1)$. Or,
Substitute the $x$ and $y$ values of the point and the value for $m$ into the gradient intercept form of a line $y=mx+b$ to determine $b$. Now you have $m$ and $b$ so you can write the equation of the line in the form $y=mx+b$.

In particular, to find the equation of the line through $(\\var{point_x},\\var{point_y})$ with a gradient of $\\simplify{{rise}/{run}}$ we can do one of the following:

Substitute $\\simplify{{rise}/{run}}$ and $(x_1,y_1)=(\\var{point_x},\\var{point_y})$ into the equation $y-y_1=m(x-x_1)$. This gives $\\simplify{y-{point_y}={rise}/{run}(x-{point_x})}$. Expanding the brackets we have $\\simplify[fractionnumbers]{y-{point_y}={rise}/{run}x-{rise*point_x/run}}$. Making $y$ the subject gives $y=\\simplify[fractionnumbers]{{rise/run}*x+{b}}$. Or,
Take the point $(\\var{point_x},\\var{point_y})$ as $(x,y)$ and substitute this and the gradient $m=\\simplify{{rise}/{run}}$ into the equation $y=mx+b$. This gives $\\simplify[!collectnumbers]{{point_y}={rise}/{run}*{point_x}+b}$. Solving for $b$ gives $b=\\var{b}$. Now we know both $m$ and $b$ and we can write the equation of the line $y=mx+b$ as $y=\\simplify[fractionnumbers]{{rise/run}*x+{b}}$.

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The gradient-intercept form of the line that passes through the point $(\\var{bpoint_x},\\var{bpoint_y})$ with a gradient of $\\simplify{{brise}/{brun}}$ is $y=$ [[0]].

There are two common ways to approach these questions:

Use the one-point gradient formula $y-y_1=m(x-x_1)$. Or,
Substitute the $x$ and $y$ values of the point and the value for $m$ into the gradient intercept form of a line $y=mx+b$ to determine $b$. Now you have $m$ and $b$ so you can write the equation of the line in the form $y=mx+b$.

In particular, to find the equation of the line through $(\\var{bpoint_x},\\var{bpoint_y})$ with a gradient of $\\simplify{{brise}/{brun}}$ we can do one of the following:

Substitute $\\simplify{{brise}/{brun}}$ and $(x_1,y_1)=(\\var{bpoint_x},\\var{bpoint_y})$ into the equation $y-y_1=m(x-x_1)$. This gives $\\simplify{y-{bpoint_y}={brise}/{brun}(x-{bpoint_x})}$. Expanding the brackets we have $\\simplify[fractionnumbers]{y-{bpoint_y}={brise}/{brun}x-{brise*bpoint_x/brun}}$. Making $y$ the subject gives $y=\\simplify[fractionnumbers]{{brise/brun}*x+{bb}}$. Or,
Take the point $(\\var{bpoint_x},\\var{bpoint_y})$ as $(x,y)$ and substitute this and the gradient $m=\\simplify{{brise}/{brun}}$ into the equation $y=mx+b$. This gives $\\simplify[!collectnumbers]{{bpoint_y}={brise}/{brun}*{bpoint_x}+b}$. Solving for $b$ gives $b=\\var{bb}$. Now we know both $m$ and $b$ and we can write the equation of the line $y=mx+b$ as $y=\\simplify[fractionnumbers]{{brise/brun}*x+{bb}}$.

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