Mathematics: Systems of Equations

Solving Systems of Equations
Systems of equations are a set of simultaneous equations that all use the same variables. A solution to a system of equations must be true for each equation in the system. Consistent systems are those with at least one solution. Inconsistent systems are systems of equations that have no solution.

Substitution
To solve a system of linear equations by substitution, start with the easier equation and solve for one of the variables. Express this variable in terms of the other variable.
Substitute this expression in the other equation, and solve for the other variable.

The solution should be expressed in the form (x, y).
Substitute the values into both of the original equations to check your answer.

Consider the following system of equations:

Solving the first equation for
:

Substitute this value in place of x in the second equation, and solve for y:

Plug this value for y back into the first equation to solve for
:

Check both equations if you have time:











Therefore, the solution is (9.6, 0.9).

Elimination
To solve a system of equations using elimination, begin by rewriting both equations in standard form . Check to see if the coefficients of one pair of like variables add to zero. If not, multiply one or both of the equations by a non-zero number to make one set of like variables add to zero. Add the two equations to solve for one of the variables.
Substitute this value into one of the original equations to solve for the other variable. Check your work by substituting into the other equation.

Now, let's look at solving the following system using the elimination method:

If we multiply the second equation by
, we can eliminate the y terms:

Add the equations together and solve for
:



Plug the value for

back in to either of the original equations and solve for y:












Therefore, the solution is (-4, 4).

Graphically
To solve a system of linear equations graphically, plot both equations on the same graph. The solution of the equations is the point where both lines cross. If the lines do not cross (are parallel), then there is no solution. following system of equations:

Since these equations are given in slope-intercept form, they are easy to graph; the y intercepts of the lines are

and
.

The respective slopes are 2 and –1, thus the graphs look like this:

The two lines intersect at the point , thus this is the solution to the system of equations.
Solving a system graphically is generally only practical if both coordinates of the solution are integers; otherwise the intersection will lie between gridlines on the graph and the coordinates will be difficult or impossible to determine exactly. It also helps if, as in this example, the equations are in slope-intercept form or some other form that makes them easy to graph. Otherwise, another method of solution (by substitution or elimination) is likely to be more useful.

Solving Systems of Equations Using the Trace Feature
Using the trace feature on a calculator requires that you rewrite each equation, isolating the y-variable on one side of the equal sign.  Enter both equations in the graphing calculator and plot the graphs simultaneously.  Use the trace cursor to find where the two lines cross.  Use the zoom feature if necessary to obtain more accurate results. 
Always check your answer by substituting into the original equations.  The trace method is likely to be less accurate than other methods due to the resolution of graphing calculators, but is a useful tool to provide an approximate answer.

Calculations Using Points
Sometimes you need to perform calculations using only points on a graph as input data. Using points, you can determine what the midpoint and distance are. If you know the equation for a line you can calculate the distance between the line and the point.

To find the midpoint of two points
-coordinates to get the <i>x</i>-coordinate of the midpoint, and average the <i>y</i>-coordinates to get the <i>y</i>-coordinate of the midpoint. The formula is: (x1 + x2)/2, (y1 + y2)/2.

The distance between two points is the same as the length of the hypotenuse of a right triangle with the two given points as endpoints, and the two sides of the right triangle parallel to the x-axis and y-axis, respectively. The length of the segment parallel to the x-axis is the difference between the x-coordinates of the two points. The length of the segment parallel to the y-axis is the difference between the y-coordinates of the two points.

Use the Pythagorean theorem

or

to find the distance. The formula is
.

When a line is in the format
, where A, B, and C are coefficients, you can use a point

not on the line and apply the formula

to find the distance between the line and the point
.

Using Points on a Graph following systems of equations:

(a)

(b)

P2. Find the distance and midpoint between points

and
.

P1. (a) If we multiply the first equation by 4, we can eliminate the

terms:



Add the equations together and solve for y:



Plug the value for y back in to either of the original equations and solve for
:

The solution is

(b) Solving the first equation for y:

Substitute this expression in place of y in the second equation, and solve for
:








value for

back in to






solution is

P2. Use the formulas for distance and midpoint: