Simultaneous linear equations (Systems of Equations) are the equations of two lines that intersect in the XY plane. (See Figure 1) You can have a pair of lines that are parallel (Never intersect) in the XY plane but they have no simultaneous solutions. (See Figure 2)

When I was learning to solve systems of linear equations, I wondered of what use is this? I found as I continued to study mathematics that the solution of simultaneous linear equations is the foundation for the solutions of more complicated problems in time and space: The calculation of the intersection of a torpedo and enemy ship or the intersection of a missile and an enemy plane or the point where a rescue space vehicle can dock with the International Space Station. Of course these examples require the use of more advanced math such as Calculus but it all begins with learning how to solve systems of linear equations.

The bases of solutions of systems of equations are the basic rules of algebraic equations.

1. Equals plus equals give equals.

2. Equals minus equals give equals.

3. Equals divided by equals give equals.

4. Equals multiplied by equals give equals.

5. An item or term may be substituted for an equivalent item or term.

Examples:

Equals + Equals give equals.

(A) y = x + 2

(B) -y = -3x +4

We have two equivalent y terms in both equations so adding them will eliminate the y terms.

(A) y= x + 2

(B) -y = -3x + 4

____________

0 = -2x +6 Now solve for x.

2x = 6 or x = 3

Now substitute the value of x into one of the equations to find y

(A) y = x + 2 x = 3

y = 3 + 2 Y = 5

So the solution to the systems of equations is x = 3 y = 5

You can check you answer by substituting the values in one of the equations

(C) -y = – 3x + 4 x = 3 y = 5

-5= -3(3) +4 -5 = -9 + 4 -5 = -5 Solutions checks.

Equals – Equals give Equals

(A) 2y = 3x +2

(B) y = 3x -4

Subtract Equation (B) from Equation (A)

(A) 2y = 3x + 2

(B) -y = -3x + 4

______________

y = 0 +6 y =6

Substitute the value of y into one of the equations to find x.

(A) y = 3x – 4 y = 6

6 = 3x -4 -3x = -4 -6 -3x = -10 x = 10/3

The solution is x = 10/3 and y = 6

Checking the solution

Y = 3x -4 6 = 3(10/3) -4 6 = 10 – 4 6 =6 Solution checks

Solving systems of equations by substitution

(A) y= 3x -7

(B) y = x +3

Substitute the value of y = x+3 from equation (B) into equation (A) and solve for x

(A) x+3 = 3x -7 x-3x = -7 -3 -2x = -10 x = 5

Substitute the value of x = 5 into one of the equations to find the value of y

(B) y = 5 + 3 y = 8

Checking the solution

(A) y = 3x -7 x = 5 y = 8

8 = 3(5) -7 8 = 15 -7 8=8 Solution checks

Sometimes you must manipulate one of the original equations in order to proceed with the solution.

(A) 3y = 2x +4

(B) 2y = x – 3

Multiply equation (B) by 2 to make the coefficients of the x equal.

(A) 3y = 2x + 4

(B) 2(2y = x – 3)

Then (A) 3y = 2x + 4

(B) 4y = 2x -6

Now we can subtract the equations

(A) 3y = 2x +4

(B) -4y = -2x +6

____________

-y = 0 + 10 -y = 10 y = -10

Substituting the value of y = -10 into one of the equations and solving for x

(A) 3y = 2x +4 3(-10) = 2x +4 -30 = 2x +4 -2x = 30 +4 -2x = 34 x = -17

Checking the solution

(A) 3y =2x + 4 x= -17 y = -10

3(-10) = 2(-17) + 4 -30 = -34 +4 -30 = -30 Solution Checks.

Some systems of equations may look like they have simultaneous solutions but truly do not since they represent parallel lines in the XY plane. You can tell by manipulating the equations into the slope intercept format y = mx + b; where m represents the slope of the line. If the equations are in slope intercept format

and the values of the x coefficients are equal the lines are parallel and have no simultaneous solutions.

Example:

(A) y = 2x +3

(B) y = 2x -5

We will attempt to solve the equations by subtraction

(A) y = 2x +3

(B) -y = -2x +5

____________

0 = 8 an absurd answer. Therefore lines are parallel (See Figure 2)

Practice solving simultaneous linear equations until you are proficient and the solutions of more complicated combinations, such as linear and quadratic equations and pairs of quadratic equations will be an easy step forward.

See the following articles in www.associatedcontent.com Keys to Solving Equations in Algebra I through VI