Worked Solution & Example Answer:Solve algebraically the simultaneous equations
$$x^2 + y^2 = 25$$
$$y - 3x = 13$$ - Edexcel - GCSE Maths - Question 20 - 2017 - Paper 1
Step 1
Substituting for y in terms of x
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Answer
From the second equation, we can express y in terms of x:
y = 3x + 13.
Step 2
Substituting y into the first equation
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Answer
Substituting the expression for y into the first equation gives us:
x2+(3x+13)2=25.
Step 3
Expanding the equation
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Answer
Expanding the left-hand side:
x2+(9x2+78x+169)=25.
This simplifies to:
10x2+78x+169−25=0,
which further simplifies to:
10x2+78x+144=0.
Step 4
Factoring the quadratic equation
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Answer
We will factor the quadratic equation:
10x2+78x+144.
It can be factored into:
2(5x+12)(x+6)=0.
Setting each factor to zero gives:
x = - rac{12}{5} \text{ or } x = -6.
Step 5
Finding corresponding y values
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Answer
Now substituting these x values back into the equation for y:
For x = - rac{12}{5}:
y = 3(- rac{12}{5}) + 13 = - rac{36}{5} + 13 = rac{29}{5}.
For x=−6:
y=3(−6)+13=−18+13=−5.
Thus, the solutions are:
(x,y)=(−512,529) and (−6,−5).
