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Algebra and Graphs - Simultaneous equations

Grade 11IGCSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Simultaneous equations consist of two or more equations with two or more unknowns that are solved together.

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The Method of Elimination: Multiplying equations to make the coefficients of one variable the same, then adding or subtracting to eliminate that variable.

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The Method of Substitution: Rearranging one equation to express one variable in terms of the other, then substituting it into the second equation.

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Graphical Interpretation: The solution to a pair of simultaneous equations is the coordinate point (x,y)(x, y) where their graphs intersect.

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Linear and Non-linear Systems: For IGCSE, you must be able to solve one linear and one quadratic equation, typically resulting in two pairs of solutions.

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Number of solutions: Two linear equations usually have one solution; a linear and a quadratic equation can have zero, one, or two solutions.

📐Formulae

Linear Equation Standard Form: ax+by=cax + by = c

Slope-Intercept Form (useful for substitution): y=mx+cy = mx + c

Quadratic Form (common in non-linear systems): y=ax2+bx+cy = ax^2 + bx + c

The Discriminant (to check for number of intersections): Δ=b2−4ac\Delta = b^2 - 4ac

💡Examples

Problem 1:

Solve the simultaneous equations using elimination: 3x+2y=183x + 2y = 18 and 2x−y=52x - y = 5.

Solution:

  1. Multiply the second equation by 2: 4x−2y=104x - 2y = 10.
  2. Add this to the first equation: (3x+2y)+(4x−2y)=18+10⇒7x=28⇒x=4(3x + 2y) + (4x - 2y) = 18 + 10 \Rightarrow 7x = 28 \Rightarrow x = 4.
  3. Substitute x=4x=4 into the second original equation: 2(4)−y=5⇒8−y=5⇒y=32(4) - y = 5 \Rightarrow 8 - y = 5 \Rightarrow y = 3. Result: x=4,y=3x = 4, y = 3.

Explanation:

Elimination is most efficient here because doubling the second equation creates a −2y-2y term which cancels the +2y+2y in the first equation.

Problem 2:

Solve the simultaneous equations: y=x2−x−4y = x^2 - x - 4 and y=2xy = 2x.

Solution:

  1. Set the equations equal to each other: x2−x−4=2xx^2 - x - 4 = 2x.
  2. Rearrange into a quadratic: x2−3x−4=0x^2 - 3x - 4 = 0.
  3. Factorise: (x−4)(x+1)=0(x - 4)(x + 1) = 0.
  4. Find x-values: x=4x = 4 or x=−1x = -1.
  5. Find corresponding y-values: If x=4,y=2(4)=8x=4, y=2(4)=8. If x=−1,y=2(−1)=−2x=-1, y=2(-1)=-2. Result: (4,8)(4, 8) and (−1,−2)(-1, -2).

Explanation:

When one equation is quadratic and the other is linear, use substitution. Setting the expressions for yy equal to each other allows you to solve for xx using quadratic methods (factorisation or the quadratic formula).