Intersection of Two Lines Revision Notes for Leaving Cert Mathematics

Intersection of Two Lines

What is the intersection of two lines?

The intersection of two lines is the point where two lines meet or cross each other on a coordinate plane. This point has coordinates that satisfy both line equations simultaneously.

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When two lines intersect, they meet at exactly one point (unless they are parallel, in which case they never meet).

Finding the intersection point

There are two main methods to find where two lines intersect:

Method 1: Graphical method

You can find the intersection point by:

Sketching both lines on the same coordinate grid
Identifying where the lines cross
Reading the coordinates of this crossing point

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The diagram above shows the lines $x + y = 4$ and $x + 3y = 6$ . By looking at the graph, you can see they intersect at the point $(3, 1)$ .

Method 2: Algebraic method using simultaneous equations

This method is more accurate and doesn't rely on drawing precise graphs. You solve the two equations together as a system of simultaneous equations.

Solving simultaneous equations

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Simultaneous equations are two or more equations that must be satisfied at the same time. The solution gives you the exact coordinates of the intersection point.

Step-by-step method:

Write both equations clearly
Use elimination or substitution to solve for one variable
Substitute back to find the other variable
Check your answer by substituting into both original equations

Worked examples

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Worked Example 1: Using elimination method

Find the intersection point of the lines:

$x + y = 5$
$2x - y = 4$

Solution: $x + y = 5 \quad \text{... ①}$ $2x - y = 4 \quad \text{... ②}$

Adding equations ① and ②: $(x + y) + (2x - y) = 5 + 4$ $3x = 9$ $\therefore x = 3$

Substituting $x = 3$ into equation ①: $3 + y = 5$ $\therefore y = 2$

The point of intersection is $(3, 2)$ .

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Worked Example 2: Another elimination example

Find the intersection point of the lines:

$x + y = 4$
$x + 3y = 6$

Solution: $x + y = 4 \quad \text{... ①}$ $x + 3y = 6 \quad \text{... ②}$

Subtracting equation ① from equation ②: $(x + 3y) - (x + y) = 6 - 4$ $2y = 2$ $\therefore y = 1$

Substituting $y = 1$ into equation ①: $x + 1 = 4$ $\therefore x = 3$

The point of intersection is $(3, 1)$ .

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Worked Example 3: Using substitution method

Find the intersection point of the lines:

$y = 2x + 1$
$3x + y = 11$

Solution: $y = 2x + 1 \quad \text{... ①}$ $3x + y = 11 \quad \text{... ②}$

Substituting equation ① into equation ②: $3x + (2x + 1) = 11$ $5x + 1 = 11$ $5x = 10$ $\therefore x = 2$

Substituting $x = 2$ into equation ①: $y = 2(2) + 1 = 5$

The point of intersection is $(2, 5)$ .

Key techniques for solving simultaneous equations

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Elimination method:

Add or subtract equations to eliminate one variable
Best used when coefficients are easy to work with
Look for opportunities where variables will cancel out

Substitution method:

Solve one equation for one variable
Substitute this expression into the other equation
Best used when one equation is already solved for a variable

Checking your answer

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Always verify your solution by substituting the coordinates back into both original equations. If the point is correct, both equations should be satisfied.

For example, if you found the intersection point $(3, 2)$ :

Check in first equation: Does $3 + 2 = 5$ ? ✓
Check in second equation: Does $2(3) - 2 = 4$ ? ✓

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Exam tips

Show all working clearly - marks are awarded for method even if the final answer is wrong
Check your arithmetic - simple calculation errors are common
Always verify your answer by substituting back into both equations
Be careful with signs when adding or subtracting equations
Label your equations with numbers ① and ② to keep track
State your final answer clearly in coordinate form $(x, y)$

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Key Points to Remember:

The intersection of two lines is the point where they meet
This point satisfies both line equations simultaneously
You can find it graphically by sketching or algebraically using simultaneous equations
Always check your answer by substituting back into both original equations
Two non-parallel lines will always intersect at exactly one point

Intersection of Two Lines (Leaving Cert Mathematics): Revision Notes