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Find the set of points of intersection of the graphs 𝑥 plus 𝑦 equals eight and 𝑥 squared plus 𝑦 squared is equal to 50.
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In order to solve this pair of simultaneous equations, we firstly need to consider the linear equation 𝑥 plus 𝑦 equals eight.
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Rearranging this equation by subtracting 𝑥 from both sides of the equation gives us 𝑦 equals eight minus 𝑥.
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Our next step is to substitute this new equation into equation two.
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𝑥 squared plus 𝑦 squared is equal to 50.
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This gives us 𝑥 squared plus eight minus 𝑥 squared equals 50.
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We can expand or simplify eight minus 𝑥 all squared using the FOIL method.
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Multiplying the first terms in the parentheses or brackets gives us 64.
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Multiplying the outside terms gives us negative eight 𝑥.
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Multiplying the inside terms also gives us negative eight 𝑥.
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And finally, multiplying the last terms gives us positive 𝑥 squared.
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This means that eight minus 𝑥 all squared is equal to 64 minus 16𝑥 plus 𝑥 squared.
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Putting this back into our equation gives us 𝑥 squared plus 64 minus 16𝑥 plus 𝑥 squared equals 50.
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Grouping the like terms gives us two 𝑥 squared minus 16𝑥 plus 64 equals 50.
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And finally subtracting 50 from both sides of this equation gives us two 𝑥 squared minus 16𝑥 plus 14 equals zero.
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Before we try and factorize this quadratic equation, we can divide by two, as all of the coefficients are even.
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Dividing by two gives us 𝑥 squared minus eight 𝑥 plus seven equals zero.
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Factorizing this quadratic gives us 𝑥 minus seven multiplied by 𝑥 minus one.
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Therefore, our two values of 𝑥 are 𝑥 equals seven or 𝑥 equals one.
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In order to find the corresponding values of 𝑦, we need to substitute these values of 𝑥 back into equation one.
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Eight minus seven is equal to one.
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And eight minus one is equal to seven.
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Therefore, 𝑦 could be equal to one or 𝑦 could be equal to seven.
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This means that the two points of intersection of the graphs 𝑥 plus 𝑦 equals eight and 𝑥 squared plus 𝑦 squared equals 50 are seven, one and one, seven.
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We could check these answers by substituting both of the coordinates back into the equations.