A system of linear equations in three variables consists of two or more linear equations containing the same three variables, where the highest power of each variable is 1. Geometrically, each linear equation in three variables represents a plane in a three-dimensional coordinate system with \(x\)-, \(y\)-, and \(z\)-axes. These systems can have a single unique solution, infinitely many solutions, or no solution at all, and can be solved algebraically using the elimination method or substitution method.
Compare a straight line in two variables with a plane in three variables, then explore the three possible intersections of three planes.
A linear equation in two variables has two axes, \(x\) and \(y\), and forms a straight line on a plane.
A linear equation in three variables has three axes, \(x\), \(y\), and \(z\), and forms a three-dimensional flat surface called a plane.
One solution: The three planes intersect at only one point \((x, y, z)\).
Infinite solutions: The three planes intersect along a straight line. Algebraically, solving yields \(0 = 0\).
No solution: The three planes do not intersect at any common point. Algebraically, solving leads to a contradiction such as \(0 = 12\).
Recognise the general form, the geometry of planes, and the two algebraic solving methods.
A linear equation in three variables has the form \(ax + by + cz = d\), where \(a\), \(b\), and \(c\) are not equal to zero, and \(a, b, c, d\) are constants.
A system refers to at least two linear equations containing the same set of variables.
Two variables form a straight line on a plane. Three variables form a plane in three-dimensional space.
One solution: planes meet at one point. Infinite solutions: planes meet along a line. No solution: planes do not share a common point.
Choose two pairs of equations from the three given equations.
Multiply equations so the coefficients of one chosen variable become equal or opposite.
Add or subtract to eliminate that variable, forming two equations in two variables.
Eliminate another variable to solve for the first variable.
Back-substitute into a two-variable equation to find the second variable.
Substitute both values into an original equation to find the third variable.
Choose an equation and express one variable in terms of the other two.
Substitute this expression into a second equation to express another variable in terms of one variable.
Substitute both expressions into the third equation to form one equation in one variable and solve it.
Back-substitute the numerical value into the derived expressions to find the remaining variables.
General form of a linear equation in three variables
where \(a, b, c \neq 0\) and all variables have power 1.
Problem solving strategy
Define variables, form three equations, solve, then verify.
Answer in your mind, then press "Check Answer".
Determine whether the following set of equations is a system of linear equations in three variables:\(p + 8q - 4r = 2\), \(2(p + 6r) + 7q = 0\), \(10r + p = 5q\).
When solving a system of linear equations in three variables, an algebraic step simplifies to \(0 = 8\). What does this result indicate about the geometric relationship of the planes and the number of solutions?
Form a linear equation in three variables from the statement: "Aiman bought 3 pairs of trousers, 2 shirts, and a pair of shoes for a total of RM750."
Three variables always require three planes that intersect at a single point.
Three planes can intersect along a line, yielding infinitely many solutions, or may not have any common intersection at all, yielding no solution.
An equation in three variables represents a 3D line.
A linear equation in two variables represents a straight line in a 2D plane, whereas a single linear equation in three variables represents a flat two-dimensional plane positioned in 3D space.
If an equation lacks one variable, such as \(y - 2z = 30\), it is not part of a three-variable system.
It is still part of the system. The coefficient of the missing variable \(x\) is simply 0: \(0x + y - 2z = 30\).
Objective: Identify linear equations, solution types, and key geometric concepts.
1 Which of the following equations is NOT a linear equation in three variables?
2 What is the geometric meaning when a system of linear equations in three variables has infinite solutions?
3 Geometrically, a linear equation in three variables forms a three-dimensional ______ when sketched on the \(x\)-, \(y\)-, and \(z\)-axes.
4 If an algebraic elimination between equations in a system leads to the contradiction \(0 = 12\), the system has ______ solution.
Drag each description to the correct solution type or geometric concept. On a phone, tap a description first, then tap the matching box.
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