Mathematics » Graphs and Equations » Use the Rectangular Coordinate System

# Finding Solutions to a Linear Equation

## Finding Solutions to a Linear Equation

To find a solution to a linear equation, you really can pick any number you want to substitute into the equation for $$x$$ or $$y.$$ But since you’ll need to use that number to solve for the other variable it’s a good idea to choose a number that’s easy to work with.

When the equation is in y-form, with the y by itself on one side of the equation, it is usually easier to choose values of $$x$$ and then solve for $$y$$.

## Example

Find three solutions to the equation $$y=-3x+2$$.

### Solution

We can substitute any value we want for $$x$$ or any value for $$y$$. Since the equation is in y-form, it will be easier to substitute in values of $$x$$. Let’s pick $$x=0$$, $$x=1$$, and $$x=-1$$.            Substitute the value into the equation.      Simplify.      Simplify.      Write the ordered pair. (0, 2) (1, −1) (−1, 5) Check. $$\phantom{\rule{0.03em}{0ex}}y=-3x+2$$ $$\phantom{\rule{0.7em}{0ex}}y=-3x+2$$ $$\phantom{\rule{0.04em}{0ex}}y=-3x+2$$ $$2\stackrel{?}{=}-3\cdot 0+2$$ $$-1\stackrel{?}{=}-3\cdot 1+2$$ $$5\stackrel{?}{=}-3\left(-1\right)+2$$ $$2\stackrel{?}{=}0+2$$ $$-1\stackrel{?}{=}-3+2$$ $$5\stackrel{?}{=}3+2$$ $$2=2✓$$ $$-1=-1✓$$ $$5=5✓$$

So, $$\left(0,2\right)$$, $$\left(1,-1\right)$$ and $$\left(-1,5\right)$$ are all solutions to $$y=-3x+2$$. We show them in the table below.

 $$y=-3x+2$$ $$x$$ $$y$$ $$\left(x,y\right)$$ 0 2 $$\left(0,2\right)$$ 1 $$-1$$ $$\left(1,-1\right)$$ $$-1$$ 5 $$\left(-1,5\right)$$

We have seen how using zero as one value of $$x$$ makes finding the value of $$y$$ easy. When an equation is in standard form, with both the $$x$$ and $$y$$ on the same side of the equation, it is usually easier to first find one solution when $$x=0$$ find a second solution when $$y=0$$, and then find a third solution.

## Example

Find three solutions to the equation $$3x+2y=6$$.

### Solution

We can substitute any value we want for $$x$$ or any value for $$y$$. Since the equation is in standard form, let’s pick first $$x=0$$, then $$y=0$$, and then find a third point.            Substitute the value into the equation.      Simplify.      Solve.            Write the ordered pair. (0, 3) (2, 0) $$\left(1,\frac{3}{2}\right)$$ Check. $$3x+2y=6\phantom{\rule{1.3em}{0ex}}$$ $$3x+2y=6\phantom{\rule{1.3em}{0ex}}$$ $$3x+2y=6\phantom{\rule{1.3em}{0ex}}$$ $$3\cdot 0+2\cdot 3\stackrel{?}{=}6\phantom{\rule{1.3em}{0ex}}$$ $$3\cdot 2+2\cdot 0\stackrel{?}{=}6\phantom{\rule{1.3em}{0ex}}$$ $$3\cdot 1+2\cdot \frac{3}{2}\stackrel{?}{=}6\phantom{\rule{1.3em}{0ex}}$$ $$0+6\stackrel{?}{=}6\phantom{\rule{1.3em}{0ex}}$$ $$6+0\stackrel{?}{=}6\phantom{\rule{1.3em}{0ex}}$$ $$3+3\stackrel{?}{=}6\phantom{\rule{1.3em}{0ex}}$$ $$6=6✓$$ $$6=6✓$$ $$6=6✓$$

So $$\left(0,3\right)$$, $$\left(2,0\right)$$, and $$\left(1,\frac{3}{2}\right)$$ are all solutions to the equation $$3x+2y=6$$. We can list these three solutions in the table below.

 $$3x+2y=6$$ $$x$$ $$y$$ $$\left(x,y\right)$$ 0 3 $$\left(0,3\right)$$ 2 0 $$\left(2,0\right)$$ 1 $$\frac{3}{2}$$ $$\left(1,\frac{3}{2}\right)$$

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