Download 6.1: Systems of Equations in Two Variables

6.1: Systems of Equations in Two Variables
A system of equations is a collection of two or more equations for which a common
solution is sought. For two equations in two variables, the set of ordered pairs that
satisfy both equations is the solution set of the system. Solving by graphing: The line
for each equation is graphed on the same plane. If the lines cross at one point, the
ordered pair that describes that point is the solution set; the system in independent. If
the lines are parallel, there is no solution; the system in inconsistent. If the two
lines are the same, there are infinitely many solutions; the system is dependent.
Determine whether the ordered pair is a solution to the system:
3x  4y  25

5x  2y  7
1. (3, -4);
2. (-4, -7);
Graph the following:
3. y = x + 4
3y – 5x = 6
3x  4y  16

5x  2y  6
4. 3y – 3x = 9
x – y = -1
y
y
9
9
8
8
7
7
6
6
5
5
4
4
3
3
2
2
1
-4
-3
-2
1
x
-1
1
2
3
4
5
6
7
-4
-3
-2
-1
-1
-2
-2
-3
-3
-4
-4
5. 2x – 3y = 6
2
y  x2
3
x
1
-1
2
3
4
5
6
7
3
4
5
6. x + 3y = -3
4x – 3y = 18
y
y
9
6
8
5
7
4
6
3
5
2
4
1
x
3
-4
2
-3
-2
-1
-2
-1
1
-1
1
-4
-3
x
1
2
3
4
5
6
7
-2
-1
-3
-2
-4
-3
-5
-4
-6
2
6
7
6.2 Solving by Substitution or Elimination (addition)
Solving by substitution: Solve one of the equations for one of the variables. The
value of that variable is substituted into the other equation for the same variable. The
second equation will now have only one variable and can be solved for that variable.
Once solved, the value of that variable is substituted into the first equation and solved
for the second variable. An independent system will have one ordered pair solution. In
an inconsistent system, the variables zero out and a false equality remains (0 = 5). In a
dependent system, the variables also zero out but the remaining equality is true (0 = 0).
Solve by substitution:
1. 2y = x + 6
-3x + 2y = -2
2. y = 3(x – 4)
3x – y = 12
3. y = 3x
y = 3x + 1
4. 3x – 5y = 4
y
3
x2
4
The Elimination (Addition) Method
To solve a system two of linear equations in two variables by addition, we add the two
equations together in such a way that one of the variables zeros out. This often requires
us to multiply one or both equations by some number that will cause one of the
variables to become additive inverses of each other (+6, -6). We solve the sum
equation for the remaining variable; then substitute that solution into either of the
original equations and solve for the other variable.
Solve by addition method:
5. 3x – 4y = 11
-3x + 2y = -7
6. 3x + 5y = -11
x – 2y = 11
7. 3x + 4y = -5
5x + 6y = -7
8. -6x + 6y = 17
x- y= 3
9.
3
5
x  y  27
7
9
1
2
x y7
9
7
10. 0.1x + 0.06y = 9
0.09x + 0.5y = 52.7
6.3 Solving Applications
1. Alkena and Hsu together earn $84,326 a year. If Alkena’s salary is $2,467 less than twice
Hsu’s salary, how much does each of them earn? Let a = Alkena’s salary; h = Hsu’s salary.
2. Gwen borrowed $10,000 for college expenses, part from Bank A at 7.5% and part from Bank
B at 8.3%. At the end of one year interest on the two loans totaled $796. How much did
she borrow from each bank? Let A = loan from Bank A and B = loan from Bank B.
3. Wendy has 52 coins consisting of nickels and pennies. If the total value of the coins is
$1.20, how many of each type does she have?
4. A 30% solution of acid is to be mixed with a 15% solution to get 12 liters of a 20% solution.
How many liters of each of the original solutions should be used?
5. Two cars leave Nashville, traveling in opposite directions. One is traveling at 70 mph and
the other at 65 mph. In how many hours will they be 540 miles apart?
6. One car leaves home at 9 a.m. traveling north at 60 mph. Three hours later a second car
leaves the same home following the same route as the first car, traveling at a speed of 75
mph. How far from Miami will the second car catch up with the first car?
7. A plane took 4 hours to fly reach its destination with a tail wind of 30 mph. The return trip
against the same wind (same engine speed) took 6 hours. What was the engine speed of
the plane? What was the distance flown? d = rt