Graphs and Simultaneous Equations
This topic connects algebra and geometry by showing that the solutions to simultaneous equations are the coordinates of the points where their graphs intersect. Understanding this visual link allows you to determine the number of solutions to an equation 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.