Simultaneous equations involving one linear equation and one non-linear equation consist of a pair of equations with two variables where one equation has degree 1 and the other has degree 2 or higher, or contains product terms such as \(xy\). Solving these equations means determining the pairs of values that satisfy both equations concurrently.
The intersection points of a straight line and a curve correspond directly to the algebraic pairs of solutions.
Substitution \(\rightarrow\) Quadratic Equation \(\rightarrow\) Two roots \(\rightarrow\) Substitute back to find matching pairs.
The straight line cuts the curve at two distinct points \((x_1, y_1)\) and \((x_2, y_2)\). These coordinates are the solutions.
Example from the textbook
Intersection points: \((1, 2)\) and \((4, -4)\)
Use substitution, elimination or graphical representation to solve the system.
A linear equation has all variable terms of degree 1, such as \(2x + y = 4\). A non-linear equation has at least one term with degree other than 1, such as \(x^2 - 2xy = 3\).
Express one variable as the subject of the formula from the linear equation, then substitute it into the non-linear equation to form a quadratic equation.
Multiply the linear equation by a suitable term, such as a variable or constant, then add or subtract it with the non-linear equation to eliminate a common term.
Construct a table of values, plot both equations on Cartesian coordinates, and read the points of intersection.
Understand the problem: identify dimensions, perimeter, area, volume or geometrical constraints.
Plan a strategy: define variables and form one linear equation and one non-linear equation.
Implement the strategy: solve using substitution and solve the resulting quadratic equation.
Make a conclusion: check feasibility in context and reject negative physical values.
Standard quadratic equation after substitution
Solve using factorisation, the quadratic formula, or completing the square.
Answer in your mind, then press "Check Answer".
Solve the simultaneous equations using the substitution method: \(x - 3y + 4 = 0\) and \(x^2 + xy - 40 = 0\).
Find the points of intersection between the straight line \(2x + y = 3\) and the curve \(\dfrac{x}{y} - \dfrac{2y}{x} = 1\).
A rectangular plank has an area of \(72\text{ cm}^2\) and a perimeter of \(34\text{ cm}\). Find the length and width of the plank.
Simultaneous equations can only be solved using the substitution method.
While substitution is the most common and versatile, the elimination method and graphical representation method are also valid approaches taught in the textbook.
Non-linear equations always have variables raised to powers greater than 1.
Terms containing the product of two variables, such as \(xy\), or reciprocal variables, such as \(\dfrac{1}{x}\), are also non-linear because their degree is not 1.
A system of one linear and one non-linear equation always yields two valid real-world solutions.
While the quadratic equation may mathematically yield two roots, in real-world contexts, negative roots for physical quantities like dimensions, time, or perimeter must be ignored.
Objective: Identify non-linear equations, interpret intersection points, and match key methods and terms.
1 Which of the following equations is a non-linear equation?
2 What do the coordinates of the intersection points between the graphs of a linear equation and a non-linear equation represent?
3 In the substitution method, one variable is made the ________ of the formula from the linear equation before being substituted into the non-linear equation.
4 When solving contextual geometric problems, any negative value obtained for a length or dimension must be ________ because physical lengths cannot be negative.
Drag each term to the correct meaning. On a phone, tap a term first, then tap the matching box.
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