Crack the Code: Pairs of Linear Equations
Crack the Code: Pairs of Linear Equations
Items in this lesson
Crack the Code: Pairs of Linear Equations
What do you already know about pairs of linear equations?
What is a Linear Equation?
A linear equation in two variables is an equation of the form ax + by + c = 0, where x and y are variables and a, b, c are constants.
Learning Objective
At the end of the lesson you will be able to identify, represent, and solve pairs of linear equations in two variables, and interpret their solutions graphically and algebraically.
Understanding Pairs
A pair of linear equations is two linear equations with the same variables. We seek values for x and y that satisfy both equations simultaneously.
General Form of Pairs
a2x + b2y + c2 = 0
Second Equation
a1x + b1y + c1 = 0
First Equation
Graphical Representation
Each linear equation graphs as a straight line. The solution of the pair is where the lines intersect on the coordinate plane.
Types of Solutions
Pairs of linear equations may have one solution (intersecting lines), no solution (parallel lines), or infinitely many solutions (coincident lines).
Unique Solution
If the lines intersect at one point, the pair has a unique solution: the coordinates of the intersection point.
No Solution
If the lines are parallel and never intersect, the pair has no solution.
Infinitely Many Solutions
If the lines overlap (are coincident), every point on the line is a solution.
Checking Consistency
A pair is consistent if it has at least one solution and inconsistent if it has none.
Consistency Conditions
No/Infinite Solutions
a1/a2 ≠ b1/b2
a1/a2 = b1/b2 ≠ c1/c2 (no solution) a1/a2 = b1/b2 = c1/c2 (infinite solutions)
Unique Solution
Solving by Substitution
Express one variable in terms of the other from one equation. Substitute into the second to solve for both variables.
Solving by Elimination
Multiply equations as needed, add or subtract to eliminate one variable. Then solve for the remaining variable.
Solving by Graphing
Plot both equations as straight lines on a graph. The intersection point(s) show the solution(s).
Real-life Applications
Pairs of linear equations help solve real-world problems like budgeting, mixtures, and comparing rates.
Worked Example 1
John buys 2 pens and 3 notebooks for £10, and 1 pen and 2 notebooks for £6. Find the cost of each item.
Practice: Try This!
Solve: 3x + 4y = 14 5x - 2y = 8
Key Tips
• Write equations clearly • Check solutions by substituting • Graph for a visual check
Reflect
How would you explain what a solution to a pair of linear equations means in your own words?
Write down 3 things you learned in this lesson.
Write down 2 things you want to know more about.
Ask 1 question about something you haven't quite understood yet.
