2.7.2 System of quadratic simultaneous equations
Two or more equations that share variables raised to powers of two such as \( x^2 \) and \( y^2 \) are said to be quadratic simultaneous equations.
To solve linear simultaneous equations, we may use
Substitution method
Graphically
Substitution method:
Solve for one of the variables from linear equation
Substitute the resulting expression in the quadratic equation
Expand and solve this quadratic equation in one variable
Substitute each solution in the linear equation to get the value of other variable
Worked example:
Solve the simultaneous equations.
You must show all your working.
\( x=7-3y \)
\( x^2-y^2=39 \)
Number the equations
\( x=7-3y \) \( (1) \)
\( x^2-y^2=39 \) \( (2) \)
Substitute \( x=7-3y \) in \( (2) \)
\( (7-3y)^2-y^2=39 \)
Expand, simplify and rearrange to form a quadratic equation
\( (7^2+(3y)^2-2×7×3y)-y^2=39 \)
\( 49+9y^2-42y)-y^2=39 \)
\( 8y^2-42y+49-39=0 \)
\( 8y^2-42y+10=0 \)
\( 4y^2-21y+5=0 \)
Solve to find values of \( y \)
\( 4y^2-20y-y+5=0 \)
\( 4y(y-5)-1(y-5)=0 \)
\( (4y-1)(y-5)=0 \)
\( 4y-1=0 \) , \( y-5=0 \)
\( y=0.25, y=5 \)
Find the corresponding \( x \) values
when \( y=0.25\) , \( x=7-3×0.25=6.25 \)
when \( y=5 \), \( x=7-3×5=-8 \)
\( x=6.25 \), \( y=0.25 \)
\( x=-8, y=5 \)
Graphically:
Rearrange linear equation in the form \( y=mx+c \)
Plot both equations on the same set of axes by using a table of values
Determine where the lines intersect
The x and y coordinates of the point of intersection are the \( x \) and \( y \) solutions to the simultaneous equations
Worked example:
Solve the simultaneous equations.
You must show all your working.
\( y=x^2+3x+1 \)
\( y=2x+1 \)
Number the equations
\( y=x^2+3x+1 \) \( (1) \)
\( y=2x+1 \) \( (2) \)
Plot both equations on the graph
Point of intersections are \( (0,1) \) and \( (-1,-1) \)
So, the solutions of the simultaneous equations are
\( x=0 \), \( y=1 \)
\( x=-1, y=-1 \)
Test yourself:
Question 1:
Solve the simultaneous equations.
You must show all your working.
\( y=5x^2+4x-19 \)
\( y=4x+1 \)
[5]
Question 2:
The straight line \( y=3x+2 \) intersects the curve \( y=2x^2+7x-11 \) at two points.
Find the coordinates of these two points.
Give your answer correct to two decimal places.
[6]
Question 3:
The graphs of \( y=x-12 \) and \( y=12x+1 \) intersect at \( A \) and \( B \).
Find the length of \( AB \).
[6]
Question 4:
Solve the simultaneous equations.
You must show all your working.
\( y=4-x \)
\( x^2+2y^2=67 \)
[6]