Introduction to Solving Pairs of Equations
The solutions of pairs of equations can be envisioned and approximated by graphing both of the equations on the same set of coordinates. Solutions appear as intersection points between the plots.
A Line And A Curve
Suppose you are given two equations in two variables, such as x and y , and are told to solve for values of x and y that satisfy both equations. Such equations are called simultaneous equations . Here is an example:
y = x ^{2} + 2 x + 1
y = − x + 1
These equations are graphed in Fig. 6-12. The graph of the first equation is a parabola, and the graph of the second equation is a straight line. The line crosses the parabola at two points, indicating that there are two solutions of this set of simultaneous equations. The coordinates of the points, corresponding to the solutions, can be estimated from the graph. It appears that they are approximately:
( x _{1} , y _{1} ) = (−3, 4)
( x _{2} , y _{2} ) = (0, 4)
You can solve the pair of equations using plain algebra, and determine the solutions exactly.
Fig. 6-12 . Graphs of two equations, showing solutions as intersection points.
Another Line And Curve
Here is another pair of “two-by-two” equations (two equations in two variables) that can be approximately solved by graphing:
y = −2 x ^{2} + 4 x − 5
y = −2 x − 5
These equations are graphed in Fig. 6-13. Again, the graph of the first equation is a parabola, and the graph of the second equation is a straight line. The line crosses the parabola at two points, indicating that there are two solutions. The coordinates of the points, corresponding to the solutions, appear to be approximately:
( x _{1} , y _{1} ) = (3, −11)
( x _{2} , y _{2} ) = (0, −5)
Again, if you want, you can go ahead and solve these equations using algebra, and find the values exactly.
Fig. 6-13 . Another example of equation solutions shown as the intersection points of their graphs.
How Many Solutions?
Graphing simultaneous equations can reveal general things about them, but should not be relied upon to provide exact solutions. In real-life scientific applications, graphs rarely show exact solutions unless they are so labeled and represent theoretical ideals.
A graph can reveal that a pair of equations has two or more solutions, or only one solution, or no solutions at all. Solutions to pairs of equations always show up as intersection points on their graphs. If there are n intersection points between the curves representing two equations, then there are n solutions to the pair of simultaneous equations.
If a pair of equations is complicated, or if the graphs are the results of experiments, you’ll occasionally run into situations where you can’t use algebra to solve them. Then graphs, with the aid of computer programs to closely approximate the points of intersection between graphs, offer a good means of solving simultaneous equations.
Sometimes you’ll want to see if a set of more than two equations in x and y has any solutions shared by them all. It is common for one or more pairs of a large set of equations to have some solutions; this is shown by points where any two of their graphs intersect. But it’s unusual for a set of three or more equations in x and y to have any solutions when considered all together (that is, simultaneously). For that to be the case, there must be at least one point that all of the graphs have in common.
Solving Pairs of Equations Practice Problems
PROBLEM 1
Using the Cartesian plane to plot their graphs, what can be said about the solutions to the simultaneous equations y = x + 3 and ( x − 1) ^{2} + ( y + 2) ^{2} = 9?
SOLUTION 1
The graphs of these equations are shown in Fig. 6-14. The equation y = x + 3 has a graph that is a straight line, ramping up toward the right with slope equal to 1, and intersecting the y axis at (0,3). The equation ( x − 1) ^{2} + ( y + 2) ^{2} = 9 has a graph that is a circle whose radius is 3 units, centered at the point (1, −2). It is apparent that this line and circle do not intersect. This means that there exist no solutions to this pair of simultaneous equations.
Fig. 6-14 . Illustration for Problem 1.
PROBLEM 2
Using the Cartesian plane to plot their graphs, what can be said about the solutions to the simultaneous equations y = 1 and ( x − 1) ^{2} + ( y + 2) ^{2} = 9?
SOLUTION 2
The graphs of these equations are shown in Fig. 6-15. The equation y = 1 has a graph that is a horizontal straight line intersecting the y axis at (0,1). The equation ( x − 1) ^{2} + ( y + 2) ^{2} = 9 has a graph that is a circle whose radius is 3 units, centered at the point (1, −2). It appears from the graph that the equations have a single common solution denoted by the point (1, 1), indicating that x = 1 and y = 1.
Fig. 6-15 . Illustration for Problem 2
Let’s use algebra to solve the equations and see if the graph tells us the true story. Substituting 1 for y in the equation of a circle (because one of the equations tells us that y = 1), we get a single equation in a single variable:
( x − 1) ^{2} + (1 + 2) ^{2} = 9
( x − 1) ^{2} + 3 ^{2} = 9
( x − 1) ^{2} + 9 = 9
( x − 1) ^{2} = 0
x − 1 =0
x = 1
It checks out. There is only one solution to this pair of simultaneous equations, and that is x = 1 and y = 1, denoted by the point (1, 1).
Practice problems for these concepts can be found at: The Cartesian Plane Practice Test.