Math 20P   Slope 1   Slope 2   Linear Equalities 1   Linear Equalities 2   Inequalities 1   Inequalities 2   Nonlinear   Prerequisites

## Graphing Systems of Linear Relations in Two Variables

What to Learn

Help to Remember What to Learn

Solve systems of linear equations, in two
variables graphically.

The intersection of the graph of two linear equations is the point(s) of intersection between the two lines.

NOTE: The fastest and easiest method of graphing is using the slope-intercept method (y=mx+b).

Relation 1: y = m1x + b1
Relation 2: y = m2x + b2

There are 3 types of solutions to systems of linear relations:

• If m1m2, the lines intersect in one place.
There is one solution - the coordinates of the ordered pair satisfying both relations.
• If m1 = m2 and b1b2, the lines are parallel (there is no intersection)
There is no solution.
• If m1 = m2 and b1 = b2, the two relations define the same line (they intersect at all points). The two relatons reduce to the same relation.
There is an infinite number of solutions - all the points on the lines.

Enrichment

• If m1m2, the lines intersect in one place, there is one solution. The system is said to be consistent and independent.
• If m1 = m2 and b1b2, the lines are parallel (there is no intersection/solution). The system is said to be inconsistent.
• If m1 = m2 and b1 = b2, the two relations define the same line (they intersect at all points). The system is said to be consistent and dependent.
 Consistent - at least one solution (Intersecting Lines) Inconsistent - no solution (Parallel Lines) Independent - lines do not overlap One solution y = 3x + 2 y = 0.5x + 6 not possible Dependent - lines overlap Infinite solutions (lines overlap) y = 3x + 2 3 * [y = 3x + 2] not possible

Note: If there are no solutions, the system is inconsistent. If there is at leason one solutions, the system is either independent or dependent.

Linear Systems Interactive Activity

Focus attention on the y = mx + b form for LINE ONE and LINE TWO.

• Manipulate the m and b sliders for each line to see see the effect on the intersection(solution) of the linear system.

## Solving Systems of Linear Equations (Substitution)

Process for a system of 2 linear equations using the variables c and d:

• Isolate d in the the first equation.
• Replace d in the second equation with the isolated value of d.
• Solve for c.
• Use the value of c to determine the value of d.
• Check your work.
• Report the answer as an ordered pair (c, d)

Sample solution:

## Solving Systems of Linear Equations (Simple Elimination)

Process for a system of 2 linear equations using the variables x and y:

• Add or subtract the system of equations to eliminate one of the variables.
• Isolate the remaining variable in the sum.
• Use the isolated value to determine the value of the second variable.
• Check your work.
• Report the answer as an ordered pair (x, y)

Sample solution: In this case the system of 2 equations can be added to eliminate y.

## Solving Systems of Linear Equations (One/Two Step Elimination)

Process for a system of 2 linear equations using two variables:

• Pick a variable to eliminate. Take a close look as one choice will likely make the solution simpler.
• Decide what the common multiple is for the coefficients of the two variables.
• For 4x and 8x, the common multiple will be 8x.
• For 2a and 3a, the common multiple will be 6a.
• Determine the factor needed to multiply by to make a new equation that will have the common multiple for the variable selected.
• Create new equation(s) my multiplying by the factor(s) determined.
• Add or subtract the new system of equations to eliminate one of the variables.
• Isolate the remaining variable in the sum.
• Use the isolated value to determine the value of the second variable.
• Check your work.
• Report the answer as an ordered pair (x, y)

Sample solution 1 (One Step) In this case multiply the first equation by 2. Subtract second equation from this new equation this new equation to eliminate x.

Sample solution 2 (Two Step) In this case multiply the first equation by 2 and the second equation by 3. Subtract the fourth equation from the third equation to eliminate a.

## 3 variable - Matrix 2

Parts of this work has been adapted from a Math 20 Pure learning resource originally produced and owned by Alberta Education.

Comments to:  Jim Reed - Homepage
Started September, 1998. Copyright © 2006, 2007