Solving systems of Linear Equations in Two Variables

1. SOLVING A SYSTEM OF EQUATIONS BY SUBSTITUTION

1.1. 1. Solve either of the equations for one variable in terms of the other

1.2. 2. Substitute the expression found in step one for the into the other equation. This will result in an equation in one variable. Solve.

1.3. 3. Substitute this value from step three into the equation found in step one. Solve for the remaining variable.

1.4. 4. Check your proposed solution in both equations.

2. ELIMINATING A VARIABLE USING THE ADDITION METHOD

2.1. 1.Write Write both equations in standard form

2.2. 2.Transform Transform the equations as needed so the coefficients of one pair of variable terms are opposites.

2.3. 3. Add Add the two equations to eliminate a variable.

2.4. 4. Solve Solve the equation from step 3 for the remaining variable.

2.5. 5. Find Find the other value, by substituting the result from step 4 into either or the original equations.

2.6. 6.Check Check you values in both original equations.

3. Equations in the form 𝐴 𝑥 + 𝐵 𝑦 = 𝐶 are linear equations. The solution to these equations are ordered pairs.

4. STEPS TO SOLVING A SYSTEM OF TWO LINEAR EQUATIONS BY GRAPHING

4.1. 1. Graph the first equation.

4.2. 2. Graph the second equation on the same coordinate plane.

4.3. 3. If the lines representing the two graphs intersect at a point, determine the coordinates of their intersection. The ordered pair is the solution 1. If the two lines are parallel, there is no solution. 2. If the two lines are the same line, there are infinitely many solutions

4.4. 4. Check the solution by plugging in the ordered pair into both equations. Example: 𝑥+2𝑦=2 𝑥−2𝑦=6

5. SOLUTIONS OF GRAPHS OF LINEAR SYSTEMS IN TWO VARIABLES

5.1. 1. The two graphs intersect in a single point. Because the system has a solution, it is consistent. The equations are not equivalent, so they are independent.

5.2. 2. The two graphs Are parallel lines. Because the system has no solution, the solution set is ∅, it is inconsistent. The equations are not equivalent, so they are independent.

5.3. 3. The two graphs are the same line. Because the are the same line, every ordered pair on the line is a solution, it is consistent. The equations equivalent, so they are dependent.