<p>Each linear equation represents a straight line. The common solution of two simultaneous linear equations is the point at which their graphs meet.</p>
<ol class="num-list"><li data-list-item-id="graph-s1" data-list-item-path="graph-s1"><div class="list-item-paragraph">For each equation, choose at least three well-spaced values of \(x\) and calculate the corresponding values of \(y\). Record them in a table.</div></li><li data-list-item-id="graph-s2" data-list-item-path="graph-s2"><div class="list-item-paragraph">Plot the points accurately and draw each straight line.</div></li><li data-list-item-id="graph-s3" data-list-item-path="graph-s3"><div class="list-item-paragraph">Draw both lines on the same pair of axes. Read the coordinates of their point of intersection.</div></li></ol>
<p>If the intersection is \((a,b)\), then&nbsp;\(x=a\) and \(y=b\) satisfy both equations.</p>
<p>For lines written as&nbsp;\(y=m_1x+c_1\) and \(y=m_2x+c_2\), the possible outcomes are:</p>
<div class="content-block content-block--table"><div class="table-container"><table style="width: 100%;"><thead><tr><th style="text-align: left; --cell-justify: flex-start">Graphs</th><th style="text-align: left; --cell-justify: flex-start">Condition</th><th style="text-align: left; --cell-justify: flex-start">Solutions</th></tr></thead><tbody><tr><td style="text-align: left; --cell-justify: flex-start">Intersecting lines</td><td style="text-align: left; --cell-justify: flex-start">\(m_1\ne m_2\)</td><td style="text-align: left; --cell-justify: flex-start">One unique solution</td></tr><tr><td style="text-align: left; --cell-justify: flex-start">Distinct parallel lines</td><td style="text-align: left; --cell-justify: flex-start">\(m_1=m_2\), \(c_1\ne c_2\)</td><td style="text-align: left; --cell-justify: flex-start">No solution</td></tr><tr><td style="text-align: left; --cell-justify: flex-start">The same line</td><td style="text-align: left; --cell-justify: flex-start">\(m_1=m_2\), \(c_1=c_2\)</td><td style="text-align: left; --cell-justify: flex-start">Infinitely many solutions</td></tr></tbody></table></div></div>

Each linear equation represents a straight line. The common solution of two simultaneous linear equations is the point at which their graphs meet.

For each equation, choose at least three well-spaced values of \(x\) and calculate the corresponding values of \(y\). Record them in a table.

Plot the points accurately and draw each straight line.

Draw both lines on the same pair of axes. Read the coordinates of their point of intersection.

If the intersection is \((a,b)\), then \(x=a\) and \(y=b\) satisfy both equations.

For lines written as \(y=m_1x+c_1\) and \(y=m_2x+c_2\), the possible outcomes are:

Graphical Method

Graphical Method

Graphs	Condition	Solutions
Intersecting lines	\(m_1\ne m_2\)	One unique solution
Distinct parallel lines	\(m_1=m_2\), \(c_1\ne c_2\)	No solution
The same line	\(m_1=m_2\), \(c_1=c_2\)	Infinitely many solutions