HA - System of Linear Equations in Two Variables Lesson

 

System of Linear Equations in Two Variables

A linear equation in two variables simply means that there are two letters holding values for the equation.  One variable is determined by the value of the other variable.  Let's examine the simplest of these equations.

y = x where x and y are both variables

When x is chosen to be 1, from the equation, when x =1 is substituted, y must equal 1.

   y = 1 where x and y are both variables.  

This creates y to be the dependent variable because its value depends on the value of x, the independent variable, the variable which is chosen first.    

Rules:  Moving from one variable to two-variable equations the major rule remains the same.

The difference is that the equations are solved for the dependent variable and the independent variable is on the opposite side of the equal sign from the dependent variable and any constant.  The example with the answer found in slope-intercept form has an independent variable x and a dependent variable y.

Example

1) 3x + y = 6

Isolate the y on one side (standard format)

2)  3x + y - 3x = 6 - 3x

Subtract 3x from both sides

3) y =  6 - 3x

Add like terms

4) y =  -3x + 6

The answer in slope-intercept form

Variable Linear Equation Formats

General Form of a linear equation is ax + by + c = 0. The letters a and b are coefficients of variables x and y respectively and c is the constant. Example:   3x + y - 6 = 0.

Standard Form of a linear equation is ax + by = c. The letters a and b are coefficients of variables x and y respectively and c is the constant.   Example:   3x + y = 6.

Slope-Intercept Form of a linear equation is y = mx + b. The coefficient of y is always 1, the coefficient of x is m, the slope of the equation, and b is a constant identifying the y-intercept of the equation. Example y = -3x - 6.

Solving a System of Equations in Two Variables

A system of equations in two variables will intersect, be parallel, or be the same line in disguise. To find which happens to a system of equations, use one of the following methods.

Lines Intersect at One Point (Consistent Independent)

Lines that have one intersection point are called consistent independent. If they were written in slope-intercept form, their slopes, m, would be different.

Solving Systems of Equations Using the Addition Method

Given two equations with the same two variables.  Write the equations one under the other with the same variable lined up vertically (missing a variable, simply use zero as a placeholder, 0x or 0y.

Find the associated x value by substituting y = 6 into one of the two equations.

x + 6 = 7

x = 1

Solve by subtracting 6 from both sides

So the solution for the system of equations is (x,y) = (1,6).  The equations intersect at (1,6).

Check your work!  Put the values (x,y) = (1,6) into the original equations.

1 + 6 = 7

7 = 7 √

-(1) + 6 = 5

5 = 5 √

Substitution Method of Solving Systems of Equations

Given two equations with the same two variables. Solve one equation for one of the variables and substitute for the variable in the other equation.

Find the associated x value by substituting x = 1 into the opposite equation from the one used to solve the two equations.

-(1) + y = 7

y = 6

Solve by adding 1 to both sides

The solution for the system of equations is (x,y) = (1,6).  The equations intersect at (1,6). Equations that intersect in one place are called consistent independent.

Of course, an equation could have been solved for x = and then substitution would have taken place for x.

Solving Systems of Equations Using the Multiplication, then Addition Method

The video below will provide examples of the use of multiplication with addition for solving systems of equations.  In many cases, the equations cannot be simply added but must have a multiplier to the equation to allow a variable to add to zero for the solution to be obtained.  The big reminder is:

Thus far, we have evaluated equations that have an intersection.  What happens when there is no intersection, or the lines intersect at all points?  

Lines are Parallel (inconsistent - no solution)

The symbol for parallel is ||. When the lines are parallel, the solution provides equality that is not true.

How is this recognizable in the equations?

-2x + 3y = 8

-2x + 3y = 5

Note that if you multiply the second equation by -1 the result would be that all of the numbers have opposite signs.

-2x + 3y = 8

2x - 3y =-5

0 = 3 Not True!

All the variables cancel to zero, but a constant remains. The original equations are parallel lines. There is no solution because parallel lines never touch. These equations are called inconsistent because they have no solution.

Another way of determining if lines are parallel is to solve the equations for y =, putting the equation in y = mx + b, slope-intercept form. If the slope, m, matches in both equations, the lines are parallel, ||.

Notice that the slope m =    is the same for both equations, so the equations are parallel, there is no intersection point. Check the slope for both equations from a grid crossing to the next clear grid crossing, both lines rise 2 units and run 3 units horizontally in the positive direction.  

Lines are Perpendicular (intersect at one point creating a 90 degree angle)

If the slopes of two equations are negative reciprocals, one slope = m and the other slope = - , the lines are perpendicular, , to each other.

Perpendicular slopes mean the lines intersect at a 90 degree angle, a special case of intersecting lines.

These equations are perpendicular because their slopes are negative reciprocals when written in the y = mx + b, slope-intercept form.

 

Lines are the Same Line (consistent dependent, infinite solutions)

How is this recognizable in the equations? 

-15x + 9y = 66

- 5x + 3y = 22

Note that if you multiply the second equation by -3 the result would be that all of the numbers are opposite signs.

-15x + 9y = 66

15x - 9y = -66

0 = 0 

All the variables and constants cancel to zero. There is an infinite number of solutions if the equations are the same. All points that solve one equation also solve the other equation.

When the lines are the same line, with an infinite number of solutions, one equation is a multiple of the other. These equations are called consistent dependent because infinite solutions are the same for both equations.

Check your knowledge of systems of equations by solving the equations below. 

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