Intersecting lines and applications

Finding where two lines meet

• The point of intersection is the point that lies on both lines, so its coordinates satisfy both equations.

• Finding it means solving the two equations simultaneously.

• The two standard methods:

  • Substitution — best when at least one equation is already $y = \ldots$
  • Elimination — best when both are in general form

Solving by substitution

• Make $y$ the subject of one equation (or use it if already done).

• Substitute that expression into the other equation.

• Solve for $x$.

• Substitute back to find $y$.

• State the answer as a point, $(x, y)$.

• Worked through — where do $y = 2x + 1$ and $y = -x + 7$ meet?

  • Set them equal: $2x + 1 = -x + 7$
  • $3x = 6$, so $x = 2$
  • Substituting: $y = 2(2) + 1 = 5$
  • Intersection: $(2, 5)$

Solving by elimination

• Multiply one or both equations so that the coefficients of one variable match.

• Add or subtract to eliminate that variable.

• Solve, then substitute back.

• Worked through — where do $3x + 2y = 16$ and $x - 2y = 4$ meet?

  • The $y$ coefficients are already $+2$ and $-2$, so add the equations: $4x = 20$, giving $x = 5$
  • Substituting into the second: $5 - 2y = 4$, so $y = 0.5$
  • Intersection: $(5, 0.5)$

Always check

• Substitute the point into BOTH original equations. A point that satisfies only one is not an intersection, and this check catches every arithmetic slip.

When there is no intersection

• Parallel lines never meet, so there is no point of intersection.
• Algebraically, solving gives something impossible such as $0 = 7$.
• Say what it means: "The lines are parallel and therefore do not intersect."

When there are infinitely many

• If the two equations describe the same line, every point on it is an intersection.
• Algebraically, solving gives something always true such as $0 = 0$.
• Check by rearranging both to $y = mx + c$ — if they match exactly, they are the same line.

Applying the whole toolkit

• Most Merit and Excellence tasks combine several methods. The common jobs:

Question asks	Method
Is the triangle isosceles?	Distance — compare side lengths
Is there a right angle?	Gradients — check $m_1m_2 = -1$
Is it a parallelogram?	Midpoints of the diagonals, or gradients of opposite sides
Is it a rhombus?	Parallelogram and all sides equal
Is it a rectangle?	Parallelogram and a right angle
Where do two paths cross?	Simultaneous equations
What is the perimeter?	Distance for each side, then add

• Read the question to decide which tool it needs. That decision is much of the assessed skill.

Relating findings to the context

• At Merit, the standard requires "relating findings to a context".
• So do not stop at "$m_1m_2 = -1$". Write: "the two paths meet at a right angle, so the corner of the reserve is square".
• At Excellence, look for a generalisation — a result that holds for a whole family of cases, not just the numbers given.