Chapter 4Section 4.1

Solving Systems of Equations in Two Variables

Systems of 2 Linear Equations

A system of 2 linear equations with two variables x and y can always be written (maybe not given this way) in the following manner:

A simultaneous solution to the system of equations is a value for x and a value for y that if you substitute the values into both of the equations they are both. We use a brace ({) to show we are looking for the values of x and y that solve both equations.

We write the solution as an ordered pair of numbers. That means if x = x0 and y = y0 is the simultaneous solution of the system of equations we write the solution as (x0,y0).

Example:

-3x + 2y = 4

4x + y = 13

This has solutions of:

x = 2 and y =5

Substitute:-3(2) + 2(5) = -6 + 10 = 4 4(2) + 5 = 8 + 5 =13

Write the solution to this system as: (2,5)

We will discuss three methods to find the solution to a system of equations1. Graphing2. Substitution3. Elimination

Solving by Graphing

As the name would imply the graphs of linear equations are lines. The idea is to graph both lines on the same graph carefully. Look at the point where the two lines cross (or try to estimate it as best as you can) the x and y coordinates are the simultaneous solutions to the system of equations.

Look at the previous example:

134

423

yx

yx

2

432

423

23

xy

xy

yx

134

134

xy

yx

-1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15-1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

slope is 3/2

y-intercept is 2

slope is -4

y-intercept is 13

The problem that you run into with graphing to find the solutions is that it can be very imprecise. When the solutions involve fractions for example reading exactly where the lines cross is hard to tell on the graph. This is why we will look at two other methods algebraic methods that tell you the simultaneous solutions.

The coordinates of the point the lines cross are (2,5)

More graphing examples:

2

3

y

x

x = -3

Vertical line at -3

y = 2

Horizontal line at 2

-5 -4 -3 -2 -1 1 2 3 4 5

-5

-4

-3

-2

-1

1

2

3

4

5

Solution (-3,2)

More graphing examples:

13

826

yx

yx

43

862

826

xy

xy

yx

13

13

13

xy

xy

yx

-5 -4 -3 -2 -1 1 2 3 4 5

-5

-4

-3

-2

-1

1

2

3

4

5

slope is 3

y-intercept is -4

slope is 3

y-intercept is -1

These lines are parallel which means they do not intersect. This means there is no simultaneous solution to the system of equations. A system of equations that has no simultaneous solution we call inconsistent or no solution.

Substitution Method

This gives the simultaneous solution by changing from 2 equations with 2 variables to one equation with one variable and solving that. Then take the value you found and plug into the substituted equation to find the other value. This will give the exact values for the simultaneous solution.

134

423

yx

yx This is the first example we started with. Use substitution to solve it.

134

134

xy

yx 1. Pick one of the equations and solve it for one of the variables. (In this case we solved the 2nd equation for y.)

413423 xx 2. Substitute for the variable you solved for into the other equation. (We substituted into the 1st equation for y.)

2

2211

42611

42683

x

x

x

xx

3. Solve the equation by isolating the variable. This will give you the value for one of the variables.

513)2(4 y 4. Substitute back into the solved equation from step 1.

5,2 5. Put the two together to form the simultaneous solution.

Solve each of the systems of equations below by substitution:

2752

33

yx

yx

33

33

yx

yx

275)33(2 yy

3

3311

27611

27566

y

y

y

yy

step 1

step 2

step 3

63)3(3 xstep 4

)3,6( step 5

This is a good method to use when one of the equations is easy to solve for one of the variables

524

72

yx

yx

72

72

72

xy

xy

yx

step 1

5)72(24 xxstep 2

514

51444

xx

step 3

This is a false statement. This means the equations has no solution.

This means there is no solution to simultaneous system of equations.

The system is inconsistent.

(Note) If you get a true statement it means the lines are the same and the number of solutions is infinite.

Elimination Method

The method again find the exact values of a simultaneous solution. The idea is to multiply one or both equations by number so that if you add them together one of the variables drops out (or is eliminated). Consider the following example.

134

423

yx

yx

)2(134

423

yx

yx 1. Multiply second equation by -2. (This gives the y variable the same coefficient but opposite in sign.)

2628

423

yx

yx2. Add the equations so now the y drops out.

2

2211

x

x3. Solve for the remaining variable.4. Plug back into one of the remaining equations.42)2(3 y

5. Solve to find the other variable.5

102

426

y

y

y

)5,2( 6. Simultaneous Solution.

243

475

yx

yx

)243(5

)475(3

yx

yx 1. Multiply first equation by 3 the second by -5 so the coefficient of x is the same number but different signs.

102015

122115

yx

yx2. Add the equations together so x drops out.

2y 3. In this case y is solved for already.

4. Plug back into one of the original equations.2)2(43 x

5. Solve for x.2

63

283

x

x

x

6. Solution of the simultaneous system.)2,2(

Solve this simultaneous system

Larger Systems

In general a system of equations that has more than two variables is difficult to understand. The graphs of these equations are hard to visualize. There are some things we know about them.

We will get only one solution (unique) if the number of variables and the number of equations is the same. For example if there are 4 variables w, x, y and z and 4 equations this system can have a single solution. It might have more than one or not any though.

If the number of variables is less than the number of equations there will never be just one solution (i.e. unique). There will either be an infinite number of different solutions or none at all.

394

1253

842

zy

zx

zyxThis system might have a unique solution.

363

8542

zyx

zyxThis system will not have a unique solution.