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How many solutions are there to the simultaneous equations 𝑥 plus seven 𝑦 equals 20 and two 𝑥 plus 14𝑦 equals 40?
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Our usual method to solve a pair of linear simultaneous equations is to make either the 𝑥 or the 𝑦 coefficients the same.
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In this case, we could multiply the first equation by two or, alternatively, we could divide the second equation by two.
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Multiplying the first equation by two gives us two 𝑥 plus 14𝑦 equals 40.
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At this stage you will notice that the two equations are actually identical.
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So, initially we thought we had a pair of simultaneous equations.
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But in reality in this case, we just need to solve one equation.
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If we consider the simplified version of the equation 𝑥 plus seven 𝑦 equals 20, we can immediately see that there are many integer value solutions.
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For example, when 𝑥 equals 13 and 𝑦 equals one, 𝑥 plus seven 𝑦 equals 20, as 13 plus seven multiplied by one is equal to 20.
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The values 𝑥 equals six and 𝑦 equals two also solve the equation as six plus seven multiplied by two is equal to 20.
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We can extend this to negative solutions.
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When 𝑥 equals negative eight and 𝑦 equals four, negative eight plus seven multiplied by four equals 20, as negative eight plus 28 is equal to 20.
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The three solutions we have seen so far are all integer solutions for 𝑥 and 𝑦.
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But when we extend this to fractions or decimals, we can see that there an infinite number of solutions to the equation 𝑥 plus seven 𝑦 is equal to 20.
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As 𝑥 plus seven 𝑦 equals 20 has an infinite number of solutions, the pair of simultaneous equations in this question must also have an infinite number of solutions.
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We can extend this one step further to say that any linear equation of the form 𝑎𝑥 plus 𝑏𝑦 equals 𝑐 where 𝑎, 𝑏, and 𝑐 are constants has an infinite number of solutions.
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In terms of simultaneous equations, there will be an infinite number of solutions if and only if the two equations are identical.
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We could also have solved this equation graphically.
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Rewriting the two equations in the form 𝑦 equals 𝑚𝑥 plus 𝑐 gives us 𝑦 equals minus a seventh 𝑥 plus 20 sevenths and the second equation 𝑦 equals negative two 14th 𝑥 plus 40 14ths.
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As the two gradients are equal, the two lines would definitely be parallel.
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Furthermore, as the interceptor are also equal, the two lines will set exactly on top of one another.
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Both lines have a gradient of negative one- seventh and a 𝑦 intercept of 20 sevenths.
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This means that they will have an infinite number of solutions.