// Numbas version: exam_results_page_options {"name": "Find the equation of a line through two points - positive gradient [L3]", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variablesTest": {"condition": "\n", "maxRuns": 100}, "variables": {"yb": {"definition": "ya+random([2,4])", "name": "yb", "description": "", "templateType": "anything", "group": "Ungrouped variables"}, "xa": {"definition": "random(-4..-1)", "name": "xa", "description": "", "templateType": "anything", "group": "Ungrouped variables"}, "m": {"definition": "(ya-yb)/(xa-xb)", "name": "m", "description": "", "templateType": "anything", "group": "Ungrouped variables"}, "ya": {"definition": "random(-4..2)", "name": "ya", "description": "", "templateType": "anything", "group": "Ungrouped variables"}, "xb": {"definition": "xa+random([2,4] except -xa)", "name": "xb", "description": "", "templateType": "anything", "group": "Ungrouped variables"}, "c": {"definition": "ya-m*xa", "name": "c", "description": "", "templateType": "anything", "group": "Ungrouped variables"}}, "extensions": ["jsxgraph"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "parts": [{"showCorrectAnswer": true, "unitTests": [], "marks": 0, "gaps": [{"maxValue": "m", "showCorrectAnswer": true, "correctAnswerStyle": "plain", "showFeedbackIcon": true, "mustBeReducedPC": 0, "unitTests": [], "marks": 1, "scripts": {}, "variableReplacements": [], "customMarkingAlgorithm": "", "allowFractions": true, "extendBaseMarkingAlgorithm": true, "minValue": "m", "notationStyles": ["plain", "en", "si-en"], "mustBeReduced": false, "correctAnswerFraction": true, "variableReplacementStrategy": "originalfirst", "type": "numberentry"}], "scripts": {}, "variableReplacements": [], "customMarkingAlgorithm": "", "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "prompt": "

Calculate the gradient, $m$, of the straight line between these two points.

\n

$m=$ [[0]]

\n

", "sortAnswers": false, "variableReplacementStrategy": "originalfirst", "type": "gapfill"}, {"showCorrectAnswer": true, "unitTests": [], "marks": 0, "gaps": [{"maxValue": "c", "showCorrectAnswer": true, "correctAnswerStyle": "plain", "showFeedbackIcon": true, "mustBeReducedPC": 0, "unitTests": [], "marks": 1, "scripts": {}, "variableReplacements": [], "customMarkingAlgorithm": "", "allowFractions": false, "extendBaseMarkingAlgorithm": true, "minValue": "c", "notationStyles": ["plain", "en", "si-en"], "mustBeReduced": false, "correctAnswerFraction": false, "variableReplacementStrategy": "originalfirst", "type": "numberentry"}], "scripts": {}, "variableReplacements": [], "customMarkingAlgorithm": "", "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "prompt": "

Use this gradient and the coordinates of the points to calculate the $y$-intercept, $c$.

\n

$c=$ [[0]]

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You must input your answer in the form y = mx +c where m and c are numbers.

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Give the equation of the straight line through these points in the form $y=mx+c$. 

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$\\displaystyle y=$ [[0]]

\n

Use the graph to plot your answer and check that it goes through these points.

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We find the equation of a straight line passing through two points by finding the gradient and the $y$-intercept of the line.

\n

a)

\n

We can find the gradient ($m$) using the points $A = (x_1,y_1)=(\\var{xa},\\var{ya})$ and $B = (x_2,y_2)=(\\var{xb},\\var{yb})$.

\n

As the definition of gradient is the ratio of vertical change ($y_2-y_1$) to horizontal change ($x_2-x_1$).
The equation for gradient is,

\n

\\begin{align}
m &= \\frac{y_2-y_1}{x_2-x_1} \\\\[0.5em]
&= \\frac{\\simplify[!collectNumbers]{{yb}-{ya}}}{\\simplify[!collectNumbers]{{xb}-{xa}}} \\\\[0.5em]
&= \\frac{\\simplify[]{{yb}-{ya}}}{\\simplify{{xb}-{xa}}} \\\\[0.5em]
&= \\simplify[simplifyFractions,unitDenominator]{({yb-ya})/({xb-xa})}\\text{.}
\\end{align}

\n

b)

\n

Rearranging the equation $y=mx+c$ and substituting either of the points gives

\n

\\[c = y_1-mx_1 \\quad \\mathrm{or} \\quad c = y_2-mx_2 \\,\\text{.} \\]

\n

We can then also use this equation with the other point's coordinates to check our answer.

\n

Let's use point $A$ first:

\n

\\[
\\begin{align}
c &= y_1-mx_1 \\\\
&= \\var{ya}-\\var[fractionnumbers]{m}\\times\\var{xa} \\\\
& = \\simplify[fractionnumbers]{{ya-m*xa}}\\text{.}
\\end{align}
\\]

\n

We then check this against point $B$:

\n

\\[
\\begin{align}
y_2 &= mx_2 + c \\\\[0.5em]
&= \\simplify[fractionNumbers]{{m}{xb}+{c}} \\\\[0.5em]
&= \\var[fractionnumbers]{m*xb+c}\\text{.}
\\end{align}
\\]

\n

c)

\n

We can now substitute these values for $m$ and $c$ into $y=mx+c$  to get:

\n

\\[y=\\simplify[!noLeadingMinus,fractionNumbers,unitFactor]{{m} x+ {c}}\\text{.}\\]

\n

The green line drawn on the graph represents the above line equation.

\n

{correctPoints()}

", "metadata": {"description": "

Use two points on a line graph to calculate the gradient and $y$-intercept and hence the equation of the straight line running through both points.

\n

The answer box for the third part plots the function which allows the student to check their answer against the graph before submitting.

\n

This particular example has a positive gradient.

", "licence": "Creative Commons Attribution 4.0 International"}, "ungrouped_variables": ["xa", "xb", "ya", "yb", "m", "c"], "statement": "

In this question we will identify the equation of the straight line passing through points  $A=(\\var{xa},\\var{ya})$ and  $B=(\\var{xb},\\var{yb})$ in the form $y = mx + c$.

\n

{plotPoints()}

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