Edexcel IGCSE Math A 2.7.HD Linear–Quadratic Simultaneous Equations

Practise substituting a linear equation into a quadratic relation and finding every real coordinate solution.

Exam points

substitute the linear expression for one variable into the quadratic equation
solve the resulting quadratic and retain every valid real root
substitute each root back to produce all ordered-pair solutions

[Maximum number: 5]

Solve the simultaneous equations

\begin{aligned} x^2+y^2&=41\\ 2x+y&=3 \end{aligned}

Show clear algebraic working.

From

2x+y=3,

y=3-2x.

Substitute into $x^2+y^2=41$ :

x^2+(3-2x)^2=41

x^2+9-12x+4x^2=41

5x^2-12x-32=0

(5x+8)(x-4)=0.

x=-\frac85\quad\text{or}\quad x=4.

Using y=3-2x:

x=-\frac85\Rightarrow y=\frac{31}{5}, \qquad x=4\Rightarrow y=-5.

Final answers:

x=-\frac85,\ y=\frac{31}{5}

x=4, y=-5

M1 for substitution. M1dep for forming a three-term quadratic. M1 for solving the quadratic. M1dep for substituting to find the other variable. A1 for all four values.

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2.7.HD Linear and quadratic simultaneous equations question 1