Download IX SOLVING TWO EQUATIONS IN TWO VARIABLES

IX SOLVING TWO EQUATIONS IN TWO VARIABLES
When linear equations are graphed in the same coordinate plane, the graphs of the
equations will do one of the following :
a) intersect in exactly one point
b) intersect in no point with the graphs of the lines being parallel
c) have every point in common and the graphs of the lines will coincide.
Two equations in two variables form a system, or pair, of simultaneous equations.
The ordered pairs of numbers that satisfy both equations are called common or
simultaneous solutions. The set of all simultaneous solutions is called the solution set of
the system. When solving the system of equations :
a) If the graphs of the lines intersect in exactly one point, exactly one ordered
pair of numbers satisfies both equations.
b) If the graphs of the lines are parallel, there is no ordered pair of numbers
that satisfies both equations.
c) If the graphs of the lines coincide, every ordered pair of numbers that
satisfies the first equation will satisfy the second equation.
Three methods of solving a system of simultaneous equations are graphing,
substitution, and linear combinations. If the graphs of the two lines intersect, it may be
possible to find the point of intersection (the solution) by graphing the equation of each
line, and then by inspecting the graph to find the point of intersection. The more accurate
methods of solving a system of equations are substitution and linear combinations.
To solve a system of equations by substitution :
1) Transform one equation to isolate one of the variables.
2) Replace the isolated variable in the second equation, making this an equation
in one variable.
3) Solve the equation
4) Replace the value for the variable in the original equation, and solve for the
remaining variable
5) Check the solution in each of the original equations.
Example 1. Solve the system
2x - y = 6
by substitution.
3x + 5y = 22
1) Transform 2x - y = 6 to isolate the y
2x - y - 6 = 0
2x - 6 = y
2) Replace the y in 3x + 5y = 22 with 2x - 6
3x + 5(2x - 6) = 22
3) Solve this equation
3x  5(2 x  6)  22
3x  10 x  30  22
13x  30  30  22  30
13x  52
13x 52

13 13
x4
4) Replace the value for x in the original equation
2x  y  6
2(4)  y  6
8 y 6
8 y 8 6 8
y  2
(1)( y )  (1)(2)
y2
5) Check the solution 4, 2 in each of the original equations.
2x  y  6
and 3x  5 y  22
2(4)   2   6 and
3(4)  5(2)  22
826
66
12  10  22
22  22
and
and
To solve a system of equations by linear combinations :
1) Multiple one or both of the equations by a number which will produce
a coefficient of x or y which has the same absolute value but different signs
2) Add these equations together
3) Find the value of the remaining variable
4) Substitute the value in one of the original equations to find the value
of the remaining variable
5) Check the solution in each of the original equations
Example 2. Solve the system :
4x - y = 10
2x + 3y = -2
by linear combinations
1) Multiply the first equation 4x - y = 10 by 3. This will produce a coefficient for
each y variable which will have the same absolute value but different signs.
3 4 x  y 10 12 x  3 y  30
2) Add this equation to the second original equation, and 3) solve for x
12 x  3 y  30
2x  3y   2
14 x  28
x 2
4) Substitute x = 2 in one of the original equations to find y
4 x  y 10
4(2)  y 10
8  y  8  10  8
y 2
y2
5) Check the solution 2, -2 in each equation.
4 x  y 10
and
2x  3y   2
4(2)   2  10
and
2(2)  3(2)   2
8   2  10
and
4  6  2
8  2 10
and
 2 2
10 10