# MATH 048 by qingyunliuliu

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```									MATH 048
SIMULTANEOUS
EQUATIONS
Solving Systems of Equations Algebraically-
Elimination Method
• Solving a system of linear equations
and subtraction is easiest when the equations
are in standard form(i.e. x and y on both sides
and constant on other side).
Solving a system of equations by elimination using
Step 1: Put the equations in            Standard Form: Ax + By = C
Standard Form.
Look for variables that have the
Step 2: Determine which            same coefficient. If none then multiply one or
variable to eliminate.             both equations by an appropriate number
to make same or opposites .

Step 3: Add or subtract the
Solve for the variable.
equations.

Step 4: Plug back in to find the   Substitute the value of the variable
other variable.                         into the equation.

BOTH equations.
.        1) Solve the system using elimination

x+y=5
3x – y = 7
Step 1: Put the equations in                  They already are!
Standard Form.

The y’s have the same
Step 2: Determine which                          coefficient.
variable to eliminate.
x+ y=5
Step 3: Add or subtract the
(+) 3x – y = 7
equations.
4x = 12
x=3
1) Solve the system using                       elimination
contd.
x+y=5
3x – y = 7
x+y=5
Step 4: Plug back in to find the
other variable.                               (3) + y = 5
y=2
(3, 2)
Step 5: Check your solution.                    (3) + (2) = 5
3(3) - (2) = 7
The solution is (3, 2). What do you think the answer would
be if you solved using substitution?
Examples. Solve using Elimination

a) x  y  9      b) x  2 y  8            c) x  2 y  9
x- y 3           x-2y  4                 2 x  18  4 y

d ) m  2n  14   e) 4 y  3x  -1          f ) 9 x  4 y  -25
3n  m  18       6 y - 7 x  -36            6 x -14 y  -25

g) 2  3  4
x   y
h) 0.04 x  0.05 y  44   i) 2( x  y)  y  1
x
4   3 4
y
x  y  1000              3( x  1)  y -3
Solving Systems of Equations
• These notes show how to solve a system
algebraically using SUBSTITUTION.

• The method is easiest to use when the system
is a system of linear equations where one of
the variable’s coefficient is 1.
Solving a system of equations by substitution
Pick the easier equation. For mixed
Step 1: Solve an equation for           solving the linear equation for
one variable.                           variable is usually easier.

Put the equation solved in Step 1
Step 2: Substitute
into the other equation.

Step 3: Solve the equation.             Get the variable by itself.

Step 4: Plug back in to find the   Substitute the value of the variable
other variable.                         into the equation.

BOTH equations.
1) Solve the system using substitution

x+y=5
y=3+x
Step 1: Solve an equation for       The second equation is
one variable.                     already solved for y!
x+y=5
Step 2: Substitute
x + (3 + x) = 5
2x + 3 = 5
Step 3: Solve the equation.                 2x = 2
x=1
1) Solve the system using substitution

x+y=5
y=3+x
x+y=5
Step 4: Plug back in to find the
other variable.                          (1) + y = 5
y=4
(1, 4)
Step 5: Check your solution.               (1) + (4) = 5
(4) = 3 + (1)
The solution is (1, 4). What do you think the answer would
be if you graphed the two equations?
Examples. Solve using substitution method

a) y  3x         b) x  4 y       c) a  b  3
x  y  83        x  8 y  52      a  3b  1

d ) 4x  y  0
8x  1 y  7
4
More Examples
• Solve the following using any method.

a ) 3 p  2r  7    b) y  x
p  2r  11      3 y  3x  4
Examples - Applications of Systems
• 1) The sum of two numbers is 56. Twice the smaller number exceeds the
larger number by 22. Find the numbers.

• 2)The sum of two numbers is 50. If twice the larger is subtracted from 4
times the smaller, the result is 8. Find the numbers.

• 3) Mrs. B bought 3 cans of corn and 5 cans of tomatoes for \$1.82. The
following week, she bought2 cans of corn and 3 cans of tomatoes for
\$1.11,paying the same prices. Find the cost of a can of corn and the cost of
a can of tomatoes.

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