Linear Systems Algebraically

Algebra Final Exam
Solving Linear Systems Algebraically Review
Procedures
Substitution Method
Elimination Method
1) At least 1 equation must be = to a variable
1) Equations must line up underneath each other.
2) Box what this variable =’s, then substitute (plug in)
2) MUST have 1 variable with same coefficient, or
this part into the other equation for that variable.
3) Solve for remaining variable
4) Once variable is found, plug answer back into one
of the equations and solve for the other variable.
5) Write solutions (x,y) and check
same coefficient with opposite signs.
3) ADD or SUBTRACT the equations together to
ELIMINATE one variable.
4) Solve for remaining variable
5) Once variable is found, plug answer back into
one of the equations and solve for the other
Example:
y  3x  23
y  11  x
Substitute this boxed
part in for y in the other
equation.
(3x + 23) –11 = x (now solve for x)
variable.
6) Write solutions (x,y) and check
Example:
2(x  y  11 ) 2x + 2y =22
3x  2 y  8  3x - 2y = 8
3x + 12 = x
-3x
-3x
12 = -2x
-2 = -2
-6 = x
Now solve for y:
Check:
y – 11 = - 6
+11 = +11
y =5
5 = 3(-6)+23
5=5
Solution: (-6, 5)
Now solve for y:
6 + y = 11
-6
-6
y =5
5x = 30
5 5
x=6
Check:
3(6) – 2(5) = 8
8=8
Solution: (6, 5)
1) At the Ralph’s consession stand during a Bills game, Vinny bought 2 pretzels and 2 boxes
popcorn and it cost him $3.50. Cory bought 2 pretzels and 4 boxes of popcorn for $6.00.
How much does a box of popcorn and a pretzel cost?
2)
Two functions, y = |x - 3| and 3x + 3y = 27, are graphed on the same set of axes.
Which statement is true about the solution to the system of equations?
(1) (3,0) is the solution to the system because it satisfies the equation y = |x + 3|.
(2) (9,0) is the solution to the system because it satisfies the equation 3x + 3y = 27.
(3) (6,3) is the solution to the system because it satisfies both equations.
(4) (3,0), (9,0), and (6,3) are the solutions to the system of equations because they all
satisfy at least one of the equations.
3)
In her restaurant Charlene sells 2 eggs and a muffin for $1.80. She sells one egg with a
muffin for $1.35. At these rates, how much is she charging for an egg and a muffin?
4)
The sum of two numbers is 47, and their difference is 15. What is the larger number?
(1) 16
(3) 32
(2) 31
(4) 36
5)
What is the value of the y-coordinate of the solution to the system of equations
x + 2y = 9 and x – y = 3?
6)
Guy and Jim work at a furniture store. Guy is paid $185 per week plus 3% of his total
sales in dollars, x, which can be represented by g(x) =185 + 0.03x. Jim is paid $275 per
week plus 2.5% of his total sales in dollars, x, which can be represented by f(x) = 275 +
0.025x. Determine the value of x, in dollars, that will make their weekly pay the same.
7)
Which system of equations has the same solution as the system below?
2x + 2y = 16
3x - y = 4
(1)
2 x  2y  16
6x  2y  4
(2)
x  y  16
3x  y  4
(3)
2 x  2y  16
6x  2y  8
(4)
6x  6y  48
6x  2y  8
8)
An arborist is selling two types of trees at his store. Tree A is three feet tall and grows at
a rate of 15 inches per year. Tree B is four feet tall an grows at a rate of 10 inches per
year. Determine how many years it will take for these trees to be the same height.
9)
An animal shelter spends $2.35 per day to care for each cat and $5.50 per day to
care for each dog. Pat noticed that the shelter spent $89.50 caring for cats and dogs
on Wednesday. Pat found a record showing that there were a total of 22 cats and
dogs at the shelter on Wednesday. How many cats were at the shelter on
Wednesday?