There are many times in physics where you start with two equations and two unknowns. You can always solve for a set of problems if you have the same number of equations as you have unknowns.
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Of of the classic problems that is so common that we learn an equation that has already combined our position & time equation with our acceleration & time equation. The equation we derive assumes that you don’t know the time and one other variable. Our two equations are:
| position & time | acceleration & time |
|---|---|
| Δx = vi ∙ t + ½ ∙ a ∙ t2 | v = a ∙ t + vi |
TI 83 or TI 4 series calculators
Graphing solution
Here are the steps for finding the intersection of two equations. Before you plug in the equations, they both need to be in the f(x)= function format. In the following example, the first equation is a momentum conservation equation, and the second is a kinetic energy conservation equation. Both are rearranged so x represents the final velocity of one object, and y represents the final velocity of the other.
In the Y= list, type in your two equations. Make sure you have the same variable identified as x and y in both equations. 
In the Zoom menu, select ZStandard. 
The scale now will likely show both your equation lines, as well as their points of intersection. If not, you may have to Zoom out some. 
Click the Calc key (2nd TRACE), then select 5 for “intersection” (of two lines). 
Look for the cursor on one of the equation lines (the line equation will show up in the upper left corner). If this is one of your lines, click Enter. If not, use the arrow keys until one of your lines is marked. 
If needed, use the arrow keys to get to a place on your second equation’s line, then press Enter. 
Use the arrow keys to move the cursor close to the predicted intersection (if there are two intersections), then click Enter. 
The final screen shows the value of X and Y when the two equations meet.