Discussion with Examples

* What is a linear equation in two variables?

A linear equation in two variables is a first-degree equation that can be put in the form:

Ax + By = C Where A, B, and C are real numbers, A and B are not 0, and x and y are the two variables

e.g. x + y = 10, 2x + 5y = 15

* What is the solution of a linear equation in two variables?

A solution of linear equation in two variables requires two values, one for each variable, which will satisfy both equations. These two variables can be represented by an ordered pair (a,b).

* How do you decide whether an ordered pair satisfies an equation in two variables?

An ordered pair (a,b) can belongs to an equation in two variables if a true statement results when a and b are substituted for x and y.

e.g. (3,2); 2x + 3y = 12

To see whether (3,2) is a solution of the equation 2x + 3y = 12, we substitute 3 for x and 2 for y in the given equation..

2x + 3y = 12 2(3) + 3(2) = 12 ?

Let x = 3; let y = 2

6 + 6 = 12 ? 12 = 12 True

The result is true, so (3,2) satisfies 2x + 3y = 12

* What are the three kinds of the systems of equations?

To illustrate what the three kinds of the systems are, let us observe these graphs of a linear system:

1) x – y = 4, x + y = 2

2) 5x – 2y = 10, 5x – 2y = 6

3) x + y = 2, 2x + 2y = 41
1) In the first graph, the lines intersect at a single point. In this case, it has only one solution. This is a consistent system.

2) In the second graph, the lines are parallel to each other. Therefore, the lines do not intersect at any point and sp the system has no solution. This is an inconsistent system.

3) In the third graph, the lines are identical; thus, they overlap and end up being the same line. In this case, the system has an infinite number of solutions since all the solutions of one equation are also solutions of the other equation.

* How do you determine whether a system is dependent, consistent, or inconsistent?

In general, the system of equations Ax + By = C and ax + by = c is:

1) consistent if A/a is not equal to B/b and B/b is not equal to C/c e.g. Consider the consistent system (x – y = 4, x + y = 2) given in the sample graph. Substituting the coefficients as stated in the formula above, 1/1 is not equal to -1/1 and -1/1 is not equal to 4/2, proving that it is indeed a consistent system.

2) inconsistent if A/a is equal to B/b and B/b is not equal to C/c e.g. Consider the inconsistent system (5x – 2y = 10, 5x – 2y = 6) given in the sample graph. Substituting the coefficients as stated in the formula above, 5/5 is equal to -2/-2 and -2/-2 is not equal to 10/6, proving that it is indeed an inconsistent system

3) dependent if A/a is equal to B/b and B/b is equal to C/c e.g. Consider the inconsistent system (x + y = 2, 2x + 2y = 4) given in the sample graph.
Substituting the coefficients as stated in the formula above, 1/2 is equal to 1/2 and 1/2 is equal to 1/2, proving that it is indeed a dependent system

* How do you solve for the solution of systems of linear equations?

There are three ways to solve systems of linear equations in two variables:

1) graphing

2) elimination method

3) substitution method

* How do you use graphing to solve for the solution of systems on linear equations?

To solve for the solution of systems on linear equations using graphing, find the x and y intercepts of each solution. The x-intercept is found by taking y = 0 and then solving for the value of x. The y-intercept is found by taking x = 0 and solving for the value of y. The x-intercept (a,0) and y-intercept (0,b) will serve as the two points to create the line.

e.g. Solve the system of equation by graphing:

In x + y = 3 and x- y = 1, you will get (respectively):

x - intercept - (3,0); y - intercept - (0,3)

x - intercept - (1,0); y - intercept - (0,-1)

Therefore, (3, 0) and (0, 3) are on the line x + y = 3 and (1, 0) and (0, -1) are on the line x – y = 1. Plotting these points and drawing the line that contains each pair we obtain this graph:

Conclusion: Since the lines intersect at (2,1), the solution is (2,1)

* How do you use the substitution method to solve for the solution of systems on linear equations?

To solve for the solution of systems on linear equations using the substitution method, follow these steps:

1) Simplify if needed. This involves removing parentheses and fractions

2) Solve one equation for either variable.

3) Substitute what you get for step 2 into the other equation.

4) Solve for the remaining variable.

e.g. Solve the system of graphing by the substitution method:

Step 1: It is already simplified

Step 2: Solving for the second equation

Step 3 and 4: Substitute the expression y + 1 for x into the first equation and solve for y:

Step 5: Plug in 3 for y into the equation in step 2 to find the value of x.

Conclusion: (4, 3) is the solution

* How do you use the elimination method to solve for the solution of systems on linear equations?

To solve for the solution of systems on linear equations using the elimination method, follow these steps:

1) Simplify and put both equations in the form Ax + By = C if needed.

2) Multiply one or both equations by a number that will create opposite coefficients for either x or y if needed.

3) Add equations.

4) Solve for second variable

5) Solve for remaining variable.

e.g. Solve the system of graphing by the elimination method:

Step 1: Multiplying each equation by it's respective LCD we get:

Step 2: Multiplying the first equation by 5 and the second equation by -2 we get:

Step 3:

Step 4:

Step 5: I choose to plug in 10 for x into the first simplified equation (found in step 1) to find the value of y. Conclusion: The solution is (10, 24)