Solving a pair of linear equations

What a pair of simultaneous equations means

• Simultaneous means "at the same time". A solution must satisfy both equations, not one of them.
• One linear equation in two unknowns has infinitely many solutions — every point on its line.
• Two equations pin it down: the solution is where the two lines cross.
• A solution is a pair of values, written as $x = \dots$, $y = \dots$ or as a point $(x, y)$. Half an answer is not an answer.

The substitution method

• Best when one equation already has a variable on its own, or is easy to rearrange.

• The steps:

  • Rearrange one equation to make a variable the subject.
  • Substitute that expression into the other equation — never back into the one it came from.
  • Solve the resulting equation in one unknown.
  • Substitute back to find the second unknown.
  • Check in the equation you did not use for the substitution.

• Worked through — solve $3x + 2y = 16$ and $5x - y = 18$:

  • Rearrange the second: $y = 5x - 18$
  • Substitute: $3x + 2(5x - 18) = 16$
  • Expand: $3x + 10x - 36 = 16$
  • Simplify: $13x = 52$, so $x = 4$
  • Back-substitute: $y = 5(4) - 18 = 2$
  • Check in the first: $3(4) + 2(2) = 16$ ✓

• Substituting back into the same equation always gives $0 = 0$. If that happens, you have used the wrong equation.

The elimination method

• Best when both equations are in the form $ax + by = c$.

• The steps:

  • Multiply one or both equations so that the coefficients of one variable match in size.
  • Add the equations if those coefficients have opposite signs; subtract if they have the same sign.
  • Solve for the remaining variable, then substitute back.

• Worked through — solve $4x + 3y = 25$ and $2x - 5y = -11$:

  • Multiply the second by 2: $4x - 10y = -22$
  • Subtract from the first: $13y = 47$…

  Setting that out properly:

  • $$(4x + 3y) - (4x - 10y) = 25 - (-22)$$
  • $13y = 47$, so $y = \dfrac{47}{13}$

• Subtracting a negative is where elimination goes wrong. Write the subtraction out in full rather than doing it in your head.

Which method to use

• The last row is the one that matters for this standard. Elimination cannot handle a squared term, so every non-linear pair is solved by substitution.

The three possible outcomes

• One solution — the lines cross once. Almost every context question is this case.
• No solution — the lines are parallel: same gradient, different intercept. Solving produces a false statement such as $0 = 7$.
• Infinitely many solutions — the lines are the same line. Solving produces a true statement such as $0 = 0$.

• $0 = 0$ and $0 = 7$ look similar and mean opposite things. A true statement means every point on the line works; a false statement means nothing works.

Connecting the representations

• The same system can be shown four ways, and a good answer moves between them:
  • In words — "two coffees and three muffins cost $21"
  • As equations — $2c + 3m = 21$
  • As a graph — two lines, crossing at the solution
  • As a table — pairs of values that satisfy each equation
• The intersection point on the graph and the algebraic solution are the same thing. Saying so explicitly is exactly the "connecting representations" that EN4 asks for.
• Graphs are for showing and checking, algebra is for finding. A reading from a graph is only as accurate as the drawing, so the algebra is the answer and the graph is the evidence.