Solving simultaneous equations - AQASimultaneous equations with linear and quadratic
Algebraic skills are required to find the values of letters within two or more equations. If two or more equations have the same variables and solutions, then they are simultaneous equations.
Simultaneous equations with one linear and one quadratic - Higher
A linear equationAn equation where the maximum power of any variable is 1. For example, 2x + 3 = 13 is a linear equation as the power of x is 1. does not contain any powers higher than 1.
A quadratic equationAn equation where the maximum power of any variable is 2. For example, y = 2x² is a quadratic equation as the power of x is 2. is an equation with the highest power of 2. For example:
\(y = x + 3\) is a linear equation and \(y = x^2 + 3x\) is a quadratic equation
Solving simultaneous equations with one linear and one quadratic
\(y = x + 3\)
\(y = x^2 + 3x\)
Substitute \(y = x + 3\) into the quadratic equation to create an equation which can be factorised and solved.
\(\mathbf{x~+~3} = x^2 + 3x\)
Rearrange the equation to get all terms on one side, so subtract \(x\) and \(-3\) from both sides:
\(x + 3 - x - 3 = x^2 + 3x - x - 3\)
\(0 = x^2 + 2x - 3\)
Factorise this equation:
\((x+3)(x-1) = 0\)
If the product of two brackets is zero, then one or both brackets must also be equal to zero.
To solve, put each bracket equal to zero.
\(\begin{array}{rcl} x + 3 & = & 0 \\ -3 && -3 \\ x & = & -3 \end{array}\)
\(\begin{array}{rcl} x - 1 & = & 0 \\ +1 && +1 \\ x & = & 1 \end{array}\)
To find the values for \(y\), substitute the two values for \(x\) into the original linear equation.
\(y = x + 3\) when \(x = -3\)
\(y = \mathbf{-3} + 3\)
\(y = 0\)
\(y = x + 3\) when \(x = 1\)
\(y = \mathbf{1} + 3\)
\(y = 4\)
The answers are now in pairs: when \(x = -3\), \(y = 0\) and when \(x = 1\), \(y = 4\)