3.2 Simultaneous Equations Involving One Linear Equation and One Non-Linear Equation
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.
Learning Objectives
- Identify and form simultaneous equations involving one linear equation and one non-linear equation from given situations.
- Solve simultaneous equations using the substitution method, the elimination method, and the graphical representation method.
- Solve quadratic equations arising from simultaneous systems using factorisation, the quadratic formula, or completing the square.
- Interpret solutions geometrically as the intersection points of the corresponding line and curve.
- Solve everyday problems involving simultaneous linear and non-linear equations, including rejecting non-feasible negative lengths or dimensions.
Geometric Representation of Linear vs Non-Linear Simultaneous Solutions
The intersection points of a straight line and a curve correspond directly to the algebraic pairs of solutions.
Algebraic Path
Substitution \(\rightarrow\) Quadratic Equation \(\rightarrow\) Two roots \(\rightarrow\) Substitute back to find matching pairs.
Geometric Meaning
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
\(2x + y = 4\) and \(y^2 - 4x = 0\)Intersection points: \((1, 2)\) and \((4, -4)\)
Short Explanation
Use substitution, elimination or graphical representation to solve the system.
1. Identifying Linear and Non-Linear Equations
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 \(x^2 - 2xy = 3\).
2. Substitution Method
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.
3. Elimination Method
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.
4. Graphical Representation Method
Construct a table of values, plot both equations on Cartesian coordinates, and read the points of intersection.
| Method | Key Idea | When to Use |
|---|---|---|
| Substitution | Replace one variable with an expression from the linear equation. | Most flexible; works for all systems. |
| Elimination | Multiply and add/subtract to cancel a term such as \(2xy\). | When a product term can be eliminated. |
| Graphical | Plot the line and curve; read intersection coordinates. | When a visual interpretation is needed. |
Steps for Solving Real-World Problems
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
\(ax^2 + bx + c = 0\)Solve using factorisation, the quadratic formula, or completing the square.
Try to Answer First
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.
Common Mistakes
- Incorrect pairing of coordinates: mismatching the roots of the quadratic equation with the wrong corresponding variable values, such as pairing \(x_1\) with \(y_2\).
- Algebraic expansion errors when substituting fractions: forgetting to square both the numerator and denominator when expanding terms like \(\left(\dfrac{4 - y}{2}\right)^2 = \dfrac{(4 - y)^2}{4}\).
- Sign errors during elimination: making sign mistakes when subtracting negative terms, such as subtracting \(-2xy\).
- Failing to reject non-physical roots: keeping negative values for lengths, widths, or dimensions in word problems where geometric quantities must be strictly positive.
- Incomplete graphical reading: estimating points of intersection inaccurately due to poor axis scaling or plotting too few points.
Concept Misunderstandings
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.
Summary
- Simultaneous equations with one linear and one non-linear equation represent the intersection between a straight line and a curve.
- They are typically solved algebraically by expressing one variable in terms of the other from the linear equation and substituting it into the non-linear equation.
- This reduces the problem to a standard single-variable quadratic equation \(ax^2 + bx + c = 0\).
- Solving this quadratic produces up to two values, which are substituted back to find the matching pair.
- In contextual problems, real-world constraints dictate whether both or only one solution is valid.
Short Activity
Objective: Identify non-linear equations, interpret intersection points, and match key methods and terms.
A. Objective Quiz
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?
B. Fill in the Blanks
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.
C. Matching / Drag and Drop
Drag each term to the correct meaning. On a phone, tap a term first, then tap the matching box.
