Graphs and Simultaneous Equations
The solutions to simultaneous equations are the coordinates of the points where their graphs intersect, connecting algebra and geometry; understanding this visual link lets you determine the number of solutions by simply sketching the corresponding graphs.
Part of the ESAT Mathematics 2 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.
Key points
- The x-coordinates of the intersection points of the graphs y = f(x) and y = g(x) are the real roots of the equation f(x) = g(x).
- The number of distinct intersection points between two graphs is equal to the number of distinct real solutions to the corresponding equation.
- If two graphs touch at a single point (i.e., they are tangent), the corresponding equation f(x) = g(x) has a repeated root at that x-value.
- Solving f(x) = g(x) algebraically gives the exact points of intersection. Sketching the graphs of y = f(x) and y = g(x) provides a quick way to find the number of solutions.
- To find the full coordinates of an intersection, solve f(x) = g(x) for x, then substitute this x-value back into either y = f(x) or y = g(x) to find the corresponding y-value.
Diagram
Formulae
f(x) = g(x) To find the x-coordinates of the points of intersection between the two graphs y = f(x) and y = g(x).
Definitions
- Simultaneous Equations
- A set of two or more equations containing the same variables. A solution is a set of variable values that satisfies all equations in the set at the same time.
- Point of Intersection
- A coordinate point (x, y) that lies on two or more graphs. The coordinates of this point satisfy the equations of all the graphs.
Worked example
The graphs of the line y = mx - 2 and the parabola y = x2 - 3x + 2 intersect at exactly one point. Find the possible values of m.
- 1
Set the two expressions for y equal to each other to find the x-coordinate of the intersection:
mx - 2 = x2 - 3x + 2 - 2
Rearrange this equation into a standard quadratic form (ax2 + bx + c = 0):
x2 - 3x - mx + 2 + 2 = 0.
- 3
Group the terms:
x2 - (3 + m)x + 4 = 0.
- 4
The prompt states there is 'exactly one point' of intersection.
This means the quadratic equation must have exactly one real root (a repeated root).
- 5
The condition for one real root is that the discriminant, b2 - 4ac, must be equal to zero.
- 6
Identify the coefficients:
a = 1, b = -(3 + m), c = 4 - 7
Substitute these into the discriminant and set it to zero:
(-(3 + m))2 - 4(1)(4) = 0.
- 8
Solve for m:
(3 + m)2 - 16 = 0, which means (3 + m)2 = 16.
- 9
Take the square root of both sides:
3 + m = 4 or 3 + m = -4 - 10
Calculate the final values:
m = 1 or m = -7
Answer: m = 1 or m = -7
Common mistakes
- ×Forgetting the condition for tangency. When asked for values of a parameter that result in one intersection point (tangency), students may fail to use the discriminant (b2 - 4ac = 0) on the resulting combined equation.
- ×Trying to solve a complex equation algebraically when a quick sketch of the two functions would immediately show how many times they intersect.
- ×Stopping after finding the x-coordinates of intersection when a question asks for the full coordinate points (x, y).
No-calculator tips
- ✓To find the number of solutions to a complex equation, rearrange it into the form f(x) = g(x), where y=f(x) and y=g(x) are simple functions you can sketch (e.g., lines, parabolas, cubics, 1/x). The number of intersections is your answer.
- ✓Master the meaning of the discriminant (b2 - 4ac) for a quadratic. A positive result means two intersections, zero means one (tangency), and negative means no intersections. This is a powerful tool for problems involving lines and parabolas.
- ✓If you solve for x and get multiple values, be sure to substitute each one back into one of the original (and preferably simpler) equations to find its corresponding y-partner.