Algebra and Functions
Simultaneous Equations and Inequalities
Pearson Edexcel International Advanced Level Mathematics
Line meets curve
- Points of intersection satisfy both equations at once.
- Substitute the linear equation into the other equation, giving a quadratic in one variable.
- Each root of the quadratic gives one point: pair each x with its own y.
- To show the x coordinates satisfy a given equation, equate, clear any fractions, and rearrange until the line reads exactly as the given equation, ending = 0.
- The number of points of intersection follows from the discriminant of that quadratic.
- Rearrange the linear equation to make y (or x) the subject.
- Substitute into the other equation.
- Collect all terms on one side as a quadratic equal to zero.
- Solve by factorising, the formula or completing the square.
- Substitute each x into the linear equation to find its y.
- State the answers as coordinate pairs.
Find, using algebra, the coordinates of the points where the curve with equation y = 2x2 − 3x − 1 meets the line with equation x + y = 3
3 − x = 2x2 − 3x − 1
2x2 − 2x − 4 = 0 ⇒ x2 − x − 2 = 0
(x − 2)(x + 1) = 0 ⇒ x = 2, −1
y = 3 − x: x = 2 gives y = 1; x = −1 gives y = 4
(2, 1) and (−1, 4)
Models with two data points
- Substitute each pair of data values into the model, giving two equations in the two constants.
- Solve the two equations simultaneously, by elimination or substitution.
- Write the model with its constants.
- Use the model to find the value asked for.
Solving inequalities
- Linear: solve as an equation; multiplying or dividing by a negative number reverses the inequality sign.
- Quadratic: rearrange to one side compared with zero, then find the critical values by solving the equation.
- Fractional, with x in the denominator: multiply both sides by x2, never by x, since x2 is positive.
| Positive x2 coefficient | Region | Answer form |
|---|---|---|
| Quadratic < 0 or ≤ 0 | between the critical values | a < x < b |
| Quadratic > 0 or ≥ 0 | outside the critical values | x < a or x > b |
- Write an inside region as one combined inequality.
- For an outside region, never write a ≥ x ≥ b; join the two parts with or.
- A sketch of the quadratic shows which region is wanted.
Find the set of values of x for which 3x(x + 1) > 2(x + 5)
3x2 + x − 10 > 0
(3x − 5)(x + 2) = 0 ⇒ x = 53, −2
> 0 with a positive x2 coefficient: outside the critical values
x < −2 or x > 53
Inequalities and graphs
- f(x) > g(x) where the graph of y = f(x) is above the graph of y = g(x).
- f(x) > 0 where the graph of y = f(x) is above the x axis.
- The critical values are the x coordinates of the points of intersection: find them algebraically, then read the region from the graph.
Regions
- Give one inequality for each boundary of the region, in terms of x and y.
- Above a boundary is y ≥; below a boundary is y ≤; an axis boundary is x ≥ 0 or y ≥ 0.
- Use strict inequalities where the boundary is not included and non-strict where it is.
- Write y ≥ or y ≤. Never write R ≥.
The region R, shown shaded, is bounded by the curve with equation y = 9 − x2, the line with equation y = x + 3 and the y axis.
Use inequalities to define the region R.
Below the curve: y ≤ 9 − x2
Above the line: y ≥ x + 3
Right of the y axis: x ≥ 0
y ≤ 9 − x2, y ≥ x + 3, x ≥ 0
These notes are part of an Excel with Osman subscription and are not available to print.
That was one page of 59
The rest of Edexcel International A-level Mathematics is written the same way
The other 58 pages of Edexcel International A-level Mathematics are written exactly like this one, with the mark scheme wording highlighted throughout. A year of the whole course is $9.99.
Subscriptions open as soon as payment processing is approved.