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Solve the simultaneous equations $$ x + y = 2 $$ $$ 4y^2 - x^2 = 11 $$ - Edexcel - A-Level Maths Pure - Question 6 - 2011 - Paper 1

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Solve the simultaneous equations $$ x + y = 2 $$ $$ 4y^2 - x^2 = 11 $$

Worked Solution & Example Answer:Solve the simultaneous equations $$ x + y = 2 $$ $$ 4y^2 - x^2 = 11 $$ - Edexcel - A-Level Maths Pure - Question 6 - 2011 - Paper 1

Step 1

1. Substitute to express y in terms of x

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Answer

From the first equation, we can express y as:

y = 2 - x$$

Step 2

2. Substitute y in the second equation

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Answer

Now substitute this expression for y into the second equation:

4(2 - x)^2 - x^2 = 11$$

Step 3

3. Expand and simplify the equation

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Answer

Expanding the left-hand side:

4(4 - 4x + x^2) - x^2 = 11$$ This simplifies to:

16 - 16x + 4x^2 - x^2 = 11$$

Combining like terms gives us:

3x^2 - 16x + 5 = 0$$

Step 4

4. Solve the quadratic equation

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Answer

We solve for x using the quadratic formula:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where a = 3, b = -16, and c = 5. Thus:

d = (-16)^2 - 4(3)(5) = 256 - 60 = 196$$

Then:

x = \frac{16 \pm \sqrt{196}}{6} = \frac{16 \pm 14}{6}$$ This gives us two values for x:

x = 5 \quad \text{or} \quad x = \frac{1}{3}$$

Step 5

5. Find corresponding y values

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Answer

Using both values of x to find y:

  1. For ( x = 5 ):
y = 2 - 5 = -3$$ 2. For \( x = \frac{1}{3} \):

y = 2 - \frac{1}{3} = \frac{5}{3}$$

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