Systems of equations on a graph

A system of equations is two equations in two unknowns, $\displaystyle x$ and $\displaystyle y$, that must hold together.
For example: the sum of two numbers is 10, and their difference is 2.
We call the larger number $\displaystyle x$ and the smaller $\displaystyle y$, and get two equations:

The curly bracket shows that the equations belong together – we look for numbers that satisfy both.

The solution of the system is an ordered pair $\displaystyle (x,y)$ that satisfies both equations. A pair is checked by substituting: put its $\displaystyle x$ and its $\displaystyle y$ into each equation.

• $\displaystyle (6,4)$: $\displaystyle {6+4}={10}$ and $\displaystyle {6-4}={2}$ – both equations hold, so $\displaystyle (6,4)$ is the solution of the system.
• $\displaystyle (7,3)$: $\displaystyle {7+3}={10}$, but $\displaystyle {7-3}={4}$, not 2 – the second equation does not hold, so $\displaystyle (7,3)$ is not a solution.

Here is the system $\displaystyle {\begin{cases}y=2x-1\\ x+y=5\end{cases}}$
Which of the pairs is the solution of the system: $\displaystyle (2,3)$, $\displaystyle (3,5)$, $\displaystyle (1,4)$?
Check each pair in both equations.

The solution of a system of linear equations is the point where its two lines meet.

• Each equation of the system is a line. We isolate $\displaystyle y$ in each:
  • in $\displaystyle {x+y}={10}$ we subtract $\displaystyle x$ from both sides: $\displaystyle {y}={-x+10}$;
  • in $\displaystyle {x-y}={2}$ we add $\displaystyle y$ to both sides and subtract 2: $\displaystyle {y}={x-2}$ (dashed).
• Every point on a line satisfies its equation.
• The meeting point is on both lines, so it satisfies both equations: $\displaystyle (6,4)$.

Here are the lines of the system $\displaystyle {\begin{cases}y=x+1\\ y=-2x+7\end{cases}}$
Mark the point where they meet on the graph.
Write the solution of the system, and check it by substituting into both equations.

The lines $\displaystyle {y}={2x+1}$ and $\displaystyle {y}={2x-3}$ (dashed) have the same slope, 2, and cross the $\displaystyle y$-axis at different points – they are parallel.

• Parallel lines never meet, so they have no point in common.
• The system $\displaystyle {\begin{cases}y=2x+1\\ y=2x-3\end{cases}}$ has no solution.
• The calculation shows it too: if $\displaystyle {2x+1}={2x-3}$, then $\displaystyle {1}={-3}$ – which is never true.

In the system $\displaystyle {\begin{cases}y=x+2\\ 2y=2x+4\end{cases}}$ the second equation is the first times 2. Divide it by 2 and you get $\displaystyle {y}={x+2}$ – the same line (the dashed one covers it).

• Every point on the line satisfies both equations: $\displaystyle (0,2)$, $\displaystyle (1,3)$, $\displaystyle (-2,0)$ …
• The system has infinitely many solutions.

The number of solutions of a system of two linear equations depends on the slopes of the two lines:

Without solving: how many solutions does each system have?
Write each equation that is not yet in the form $\displaystyle {y}={\ldots}$ in that form, and compare the slopes and the points on the $\displaystyle y$-axis.

$\displaystyle {\begin{cases}y=3x+1\\ y=3x-2\end{cases}}$

$\displaystyle {\begin{cases}y=-x+4\\ y=2x+4\end{cases}}$

$\displaystyle {\begin{cases}y=2x-6\\ 2y=4x-12\end{cases}}$

$\displaystyle {\begin{cases}x+y=5\\ y=x+5\end{cases}}$