Fine Down the equation of the following straight lines in the form of y= mx + c    M     C
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<h2>The Gradient–Intercept Form of a Straight Line Equation</h2>
<p>Using the gradient–point form, we can also derive the gradient–intercept form of the straight line equation.</p>
<p>Starting with the equation</p>
<p>\[y - y_1 = m(x - x_1)\]</p>
<p>Expand the brackets and make \(y\) the subject of the formula</p>
<p>\begin{align*} y - y_1 &amp;= mx - mx_1 \\ y &amp;= mx - mx_1 + y_1 \\ y &amp;= mx + (y_1 - mx_1) \end{align*}</p>
<p>We define constant \(c\) such that \(c={y}_{1}-m{x}_{1}\) so that we get the equation</p>
<p>\[y = mx + c\]</p>
<p>This is also called the <strong>standard form</strong> of the straight line equation.</p>
<p>Notice that when \(x=0\), we have</p>
<p>\begin{align*} y &amp;= m(0) + c \\ &amp;= c \end{align*}</p>
<p>Therefore \(c\) is the \(y\)-intercept of the straight line.</p>
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<h2>Example</h2>
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<h3>Question</h3>
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<p>Determine the equation of the straight line with gradient \(m=-2\) and passing through the point \((-1;7)\).</p>
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<h3>Slope of the line</h3>
<p>We notice that \(m &lt; 0\), therefore the graph decreases as \(x\) increases.</p>
<img alt="4801b97bb94e1995eca2a027b9952403.png" src="https://data.ulearngo.com/assets/36c5a7ab-9796-44fa-9571-ac8496887b8d"/></div>
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<h3>Write down the gradient–intercept form of straight line equation</h3>
<p>\[y =mx + c\]</p>
<p>Substitute the value of the gradient</p>
<p>\[y = -2x + c\]</p>
<p>Substitute the coordinates of the given point and find \(c\)</p>

\begin{align*} y &amp;= -2x + c \\ 7 &amp;= -2(-1) + c \\ 7 - 2 &amp;= c \\ \therefore c &amp;= 5 \end{align*}

<p>This gives the \(y\)-intercept \((0;5)\).</p>
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<h3>Write the final answer</h3>
<p>The equation of the straight line is \(y = -2x + 5\).</p>
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<p>If we are given two points on a straight line, we can also use the gradient–intercept form to determine the equation of a straight line. We solve for the two unknowns \(m\) and \(c\) using simultaneous equations — using the methods of substitution or elimination.</p>
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<h2>Example</h2>
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<h3>Question</h3>
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<p>Determine the equation of the straight line passing through the points \((-2;-7)\) and \((3;8)\).</p>
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<h3>Draw a sketch</h3>
<img alt="cab7c474ca39f69ba4dfa5e849645825.png" src="https://data.ulearngo.com/assets/0fb4788e-7e1d-400a-bb11-3beba4168cd1"/></div>
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<h3>Write down the gradient–intercept form of straight line equation</h3>
<p>\[y =mx + c\]</p>
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<h3>Substitute the coordinates of the given points</h3>

\[\begin{array}{rll} -7 &amp; = m(-2) + c &amp; \\ -7 &amp; = -2m + c &amp; \ldots (1) \\ \\ 8 &amp; = m(3) + c &amp; \\ 8 &amp; = 3m + c &amp; \ldots (2) \end{array}\]

<p>We have two equations with two unknowns; we can therefore solve using simultaneous equations.</p>
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<h3>Make the coefficient of one of the variables the same in both equations</h3>
<p>We notice that the coefficient of \(c\) in both equations is \(\text{1}\), therefore we can subtract one equation from the other to eliminate \(c\):</p>

\begin{align*} -7 &amp;= -2m + c \\ -( 8 &amp;= 3m + c ) \\ -15 &amp;= -5m \\ \therefore 3 &amp;= m \end{align*}

<p>Substitute \(m=3\) into either of the two equations and determine \(c\): \begin{align*} -7 &amp;= -2m + c \\ -7 &amp;= -2(3) + c \\ \therefore c &amp;= -1 \end{align*} or \begin{align*} 8 &amp;= 3m + c \\ 8 &amp;= 3(3) + c \\ \therefore c &amp;= -1 \end{align*}</p>
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<h3>Write the final answer</h3>
<p>The equation of the straight line is \(y = 3x - 1\).</p>
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The Gradient–Intercept Form of a Straight Line Equation

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Mathematics

Discover the principles of analytical geometry, including drawing figures on the Cartesian plane, finding the distance between two points, calculating the gradient of a line, identifying parallel and perpendicular lines, determining the midpoint of a line, solving for the equation of a straight line and circle, and more.


The Gradient–Intercept Form of a Straight Line Equation

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