Question: Solve the simultaneous equations $2y = 3x - 13$ and $5x - 6y = 23$.

Step 1. Number the equations and aim to make one variable's coefficients match:

$(1) \times 3$ turns $2y$ into $6y$:

Step 2. $(2) + (3)$ — adding kills the $-6y/+6y$ pair:

Step 3. Solve for $x$:

Step 4. Substitute back into the easier equation, (1):

Check — in the equation you did NOT use: $(2)$: $5(4) - 6\left(-\frac{1}{2}\right) = 20 + 3 = 23$. ✓

Elimination is the default when both equations arrive in ax + by = c form; rearranging one into y = … first usually signals substitution instead.

⚠ Watch out: Decide add-or-subtract by the signs of the matched terms: opposite signs ($-6y$ and $+6y$) add; same signs subtract. When multiplying an equation, multiply every term — the constant included. The final check must use the unused equation, or it proves nothing.

Question: Solve $y = 2x + 1$ and $y = x + 3$ by substitution, and state what the solution means graphically.

Step 1. Both right-hand sides equal the same $y$ — substitute one into the other:

Step 2. Solve the resulting single-variable equation:

Step 3. Feed $x = 2$ back into either original equation:

Check in the other: $y = 2 + 3 = 5$. ✓ Solution: $x = 2, y = 5$.

Step 4. Graphically, each equation is a straight line, and the solution is their point of intersection: the lines $y = 2x+1$ and $y = x+3$ cross exactly at $(2, 5)$. That is also why a pair of parallel lines (same gradient, different intercepts) gives no solution, and two copies of the same line give infinitely many.

⚠ Watch out: Substitution wants one variable already isolated — here $y$ was, for free. After finding $x$, finish the job: the answer is the pair $(x, y)$, and the graphical-meaning sentence ("the point of intersection of the two lines") is a stated mark on its own.

Question: $x = 2$, $y = -1$ is the solution of the equations $ax + by = 3$ and $ax - by = 17$. Find the values of $a$ and $b$.

Step 1. A "solution" satisfies both equations — substitute $x = 2, y = -1$ into each:

Step 2.

The unknowns have swapped roles: the new simultaneous pair is in $a$ and $b$.

Step 3. $(1) + (2)$ eliminates $b$ instantly:

Step 4. Substitute into (2):

Check. $(1)$: $2(5) - 7 = 3$. ✓ So $a = 5$, $b = 7$.

Notice the labour split: the substitution step builds the equations, and ordinary elimination finishes them — two familiar skills chained.

⚠ Watch out: The sign trap sits in Step 2: $-b(-1) = +b$ — a double negative that flips the equation's character. Substituting a known solution to create equations (rather than solve them) is a recurring exam move; recognise it whenever the question hands you $x$ and $y$ but asks for the coefficients.