Simultaneous equations

Solving pairs of linear equations by graphing, substitution and elimination, and checking the solution in both equations.

The graphical view

Two lines, one solution

The solution to the pair of equations is the point of intersection.

Example 1●●

Solve $y = 2x - 1$ and $3x + y = 14$.

$3x + (2x - 1) = 14$

Substitute y

$5x = 15$, so $x = 3$, and $y = 2(3) - 1 = 5$.

Example 2●●

Solve $2x + 3y = 12$ and $4x - 3y = 6$.

$6x = 18$, so $x = 3$.

Add the equations: 3y and −3y cancel

$2(3) + 3y = 12$, so $y = 2$.

Example 3●●●

Solve $3x + 2y = 16$ and $2x + y = 9$.

$4x + 2y = 18$

Multiply the second equation by 2

$x = 2$, then $y = 9 - 4 = 5$.

Subtract the first equation

Flashcards

Substitution works best when

One equation already says $y = \ldots$ or $x = \ldots$

Elimination

Make one pronumeral's coefficients match, then add or subtract.

Same sign coefficients

Subtract. Opposite signs: add.

Practice