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Module 6 (M6) – Algebra - Simultaneous equations

Part of MathsM6: Algebra

Before starting this guide, it may be helpful to read the guide from M1 on coordinates and graphs.

Key points

Image caption,
Simultaneous equations like 𝒚 = 2𝒙 - 1 and 𝒚 = 𝒙 + 1 can be represented graphically.

equations are two or more equations that share .
For example, \(y\) = 2\(x\) - 1 and \(y\) = \(x\) + 1 share the variables \(x\) and \(y\). They are simultaneous because the equations are solved at the same time.
Simultaneous equations can be solved algebraically or graphically.
To solve simultaneous equations graphically, it is essential to be able to draw the graph of a straight line.

Image caption,
Simultaneous equations like 𝒚 = 2𝒙 - 1 and 𝒚 = 𝒙 + 1 can be represented graphically.

Representing equations graphically

Each equation is used to create a table of values, which are then plotted as pairs on the same set of .
Graphs are often written in the form \(y\) = \(mx\) + \(c\), where \(m\) is the (how steep the line is) and \(c\) is the \(y\)-.
If the two lines cross, then the coordinates of this are the solutions to the simultaneous equations. If the lines are parallel (and therefore do not intersect), there is no solution.

Example

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Image caption,
Use the graphical method to solve the simultaneous equations.
Image caption,
Both equations are in the form 𝒚 = 𝒎𝒙 + 𝒄. For the equation 𝒚 = 𝒙 + 1, the gradient (𝒎) is 1 and the intercept (𝒄) is +1. For the equation 𝒚 = 2𝒙 – 1, the gradient (𝒎) is 2 and the intercept (𝒄) is -1
Image caption,
To solve the equations graphically, the two lines 𝒚 = 𝒙 + 1 and 𝒚 = 2𝒙 - 1 are drawn on the same diagram. The coordinates of the point of intersection of the two graphs are the solutions to the simultaneous equations. Draw a table of values for the graph 𝒚 = 𝒙 + 1. A minimum of three pairs of values are required. When 𝒙 = 0 then 𝒚 = 1 (0 + 1 = 1). When 𝒙 = 1 then 𝒚 = 2 (1 + 1 = 2). When 𝒙 = 2 then 𝒚 = 3 (2 + 1 = 3).
Image caption,
Draw a table of values for the graph 𝒚 = 2𝒙 - 1. A minimum of three pairs of values are required. When 𝒙 = 0 then 𝒚 = -1 (2 x 0 - 1 = -1). When 𝒙 = 1 then 𝒚 = 1 (2 x 1 - 1 = 1). When 𝒙 = 2 then 𝒚 = 3 (2 x 2 - 1 = 3).
Image caption,
Plot the three coordinate pairs from the table onto the axes. Extend the line. This is the graph for the equation 𝒚 = 𝒙 + 1
Image caption,
Plot the three coordinate pairs from the table onto the axes. Extend the line. This is the graph for the equation y = 2𝒙 - 1
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Draw both graphs on the same set of axes. Label each line with the correct equation.
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The two lines intersect with each other. This means there is a solution. The lines intersect at the point (2, 3). The solution to the simultaneous equations is 𝒙 = 2 and 𝒚 = 3

Question

What is the solution to the pair of equations \(y\) = \(x\) + 3 and \(y\) = 2\(x\) + 1?

The image shows a set of axes. The horizontal axis is labelled x. The values are increasing in units of two from negative two to positive six. The vertical axis is labelled y. The values are increasing in units of two from zero to ten. The functions y equals two x plus one and y equals x plus three are plotted and labelled on the axes. The line for the function y equals two x plus one is coloured blue. The line for the function y equals x plus three is coloured orange.

Rearranging equations to solve problems

To be able to solve simultaneous equations, both equations need to be in the form \(y\) = \(mx\) + \(c\)
If the equations are not in the form \(y\) = \(mx\) + \(c\), the equations need to be rearranged.

Example

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Image caption,
Use the graphical method to solve the simultaneous equations.
Image caption,
The two equations need to be in the form 𝒚 = 𝒎𝒙 + 𝒄. The equation 𝒚 = 2𝒙 is written in this form. However, 𝒚 + 2𝒙 = 4 is not and needs to be rearranged.
Image caption,
Rearrange the second equation. The two equations are now 𝒚 = 2𝒙 and 𝒚 = -2𝒙 + 4. Each of these two equations is in the form 𝒚 = 𝒎𝒙 + 𝒄 and can be plotted as a straight-line graph.
Image caption,
This graph shows 𝒚 = 2𝒙. The straight line shows the different values of 𝒙 and 𝒚 that follow the rule. For example, for the point (1, 2), the 𝒙 coordinate is 1 and the 𝒚 coordinate is 2. The point (2, 4) is another pair of values that follows the same rule. The 𝒙-coordinate is 2 and the 𝒚-coordinate is 4
Image caption,
This graph shows 𝒚 = -2𝒙 + 4. The straight line shows the different values of 𝒙 and 𝒚 that follow the rule. For example, for the point (2, 0), the 𝒙-coordinate is 2 and the 𝒚-coordinate is 0. There are lots of other pairs of points on the graph that also follow the rule.
Image caption,
To solve the simultaneous equations draw both graphs on the same axes.
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The two lines cross at the point (1, 2). This is called the point of intersection. The values of 𝒙 and 𝒚 at this single point are the solutions of the simultaneous equations. The solution to the equations 𝒚 = 2𝒙 and 𝒚 + 2𝒙 = 4 is 𝒙 = 1 and 𝒚 = 2

Question

What is the solution to the pair of equations \(y\) = \(x\) + 5 and \(y\) = 7 − \(x\)?

The image shows a set of axes. The horizontal axis is labelled x. The values are increasing in units of one from negative one to positive seven. The vertical axis is labelled y. The values are increasing in units of one from zero to seven. The functions y equals x plus five and y equals seven subtract x are plotted and labelled on the axes. The line for the function y equals x plus five is coloured orange. The line for the function y equals seven subtract x is coloured blue.

Real-life maths

The image shows a set of axes. The horizontal axis is labelled output, units. The values are increasing in units of twenty from zero to two hundred. The vertical axis is labelled cost and revenue in pounds. The values are increasing in units of four hundred from zero to two thousand. Two linear functions are plotted showing total costs and revenue. These intersect at a point where the company break even. The region between the two lines to the left represents a loss. The region between the two lines to the right represents a profit. — Image caption,
This graph shows the break-even point for a company looking at their costs and outputs.

A company will use graphs of simultaneous equations to model the potential profits it can make by selling its products at a different price. The company will have fixed costs and costs, depending on how much of something it produces.
will be made by multiplying the sale price by the number of units sold. Where the two lines on the graph intersect will be the ‘break-even’ point (the number of sales needed before the company begins to make a profit). Using a spreadsheet, the company can make financial decisions, including the profits they might make or the effect of lowering (or raising) the price of what it sells.

The image shows a set of axes. The horizontal axis is labelled output, units. The values are increasing in units of twenty from zero to two hundred. The vertical axis is labelled cost and revenue in pounds. The values are increasing in units of four hundred from zero to two thousand. Two linear functions are plotted showing total costs and revenue. These intersect at a point where the company break even. The region between the two lines to the left represents a loss. The region between the two lines to the right represents a profit. — Image caption,
This graph shows the break-even point for a company looking at their costs and outputs.

Test yourself

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