Solve SAT systems of equations with Desmos

Topic: Systems of equations
By: SolvePlot

Graph both equations without rearranging them. An intersection gives values of x and y that satisfy both equations. Read the coordinate the question asks for, then check the pair in the original system.

Original worked examples

A system asks for values that work in more than one relationship at once. On a graph, the shared points are intersections. You do not have to convert each line to slope-intercept form before entering it.

Two distinct lines can cross once, be parallel, or represent the same line. A single visible intersection is a solution; seeing no intersection in the current window is not enough to prove that none exists.

Can Desmos solve a system without substitution?

For numerical equations in $x$ and $y\text{,}$ often yes: enter one equation per row and inspect their intersection. If the question asks for an expression such as $x + y\text{,}$ first check whether one of the given equations already fixes it. Finding both coordinates could be unnecessary.

For a symbolic question about all possible coefficients, an approximate graph may not establish the claim. Algebra may be needed to prove when lines are parallel or identical.

Original example

Work through the question

The numbers $x$ and $y$ satisfy $x+y=12$ and $2x-y=3\text{.}$ What are $x$ and $y\text{?}$

Answer $(x,y)=(5,7)$

The lines x + y = 12 and 2x − y = 3 intersect at (5, 7). Illustration; follow the rows below in Desmos.

What to enter in Desmos

Use one expression row for each entry. Underscores create subscripts; the tilde ~ fits parameters.

```
x+y=12
```
Graph the first relationship exactly as written.
```
2x-y=3
```
Graph the second relationship on a separate row.

Read the result

Select the crossing point at (5, 7). The first coordinate is x = 5 and the second is y = 7. If it is outside the window, zoom out or set a window covering x from 0 to 10 and y from 0 to 15.

Check it

$5+7=12$ and $2(5)-7=3\text{,}$ so the pair satisfies both equations.

Try the entries in Desmos

The algebraic solution

Add the equations. The $y$ terms cancel, leaving $3x=15\text{.}$
Then $x=5\text{.}$ Substitute into $x+y=12$ to get $y=7\text{.}$

When is each method useful?

Use Desmos when

Graphing is a useful direct route when the equations are awkward to rearrange or one relationship is nonlinear. It also gives a quick way to check a coordinate pair.

Use algebra when

Elimination is short in this example because the y coefficients are opposites. It is also useful when an exact fraction or a general statement about parameters is required.

Common mistakes to avoid

Do not read the x-intercept of one line instead of their intersection.
Keep the minus sign in the second equation. Changing it changes the system.
If the question asks only for x, do not report the y-coordinate.
Coincident lines have infinitely many shared points. A numerical screenshot alone does not explain why the equations are equivalent.

Try one yourself

Solve $x+y=9$ and $x-y=1$

See the answer and steps

$(x,y)=(5,4)$

The graphs intersect at $\left(5,\,4\right)\text{.}$ Adding the equations gives $2x=10\text{,}$ then $y=4\text{.}$ Check: $5+4=9$ and $5-4=1\text{.}$

Desmos entries:

x+y=9
x-y=1

Calculator references

The examples above are original. These official resources explain the calculator features and test-day rules.

Desmos: graphing calculator user guide