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Lesson 5.1, page 252Systems of Linear Equations

in Two Variables

- Objective To solve systems of equations using

substitution or elimination.

Definitions

- System 2 or more equations together
- Solution of system any ordered pair that makes

all equations true - Possible Solutions one point, all points, no

points

See Example 2, page 253.

- Are the given ordered pairs solutions of the

given systems? - x - y 17 and x y -1 (8,-9)
- 3x 5y -12 and x - y 1 (-1,2)

Real World ConnectionRead Say Cheese! Pg. 252

- Do you think graphing is the only way to solve a

system? - What if the equations are not easy to graph?
- We can also ALGEBRA to solve systems by

Substituting or the process of Elimination.

Steps for using SUBSTITUTION

- Solve one equation for one variable. (Hint

Look for an equation already solved for one of

the variables or for a variable with a

coefficient of 1.) - Substitute into the other equation.
- Solve this equation to find a value for a

variable. - Substitute again to find the value of the other

variable. - Check.

See Examples 3 4, page 254-5.

- c) 2x y 6 d) x 6y - 2
- y 5x x ¾y

What if one of the equations is not easily or not

already solved for a variable?

- Use ELIMINATION!
- Elimination method to use math operations to

eliminate a variable

STEPS for ELIMINATION

- 1) Look for a variable that has OPPOSITE

coefficients. If found, ADD the two equations.

Solve, substitute, solve. - 2) If no opposite coefficients, look for

variables with the SAME coefficients. If found,

SUBTRACT one equation from the other. Then,

solve, substitute, solve. - Note Possible solutions are the same as before
- One point in common
- Infinitely many solutions
- No solution

See Example 5, page 256.

- e) x y -2 f) x 3y -5
- x y 6 x 2y 0

What if neither variable can be eliminated by

simply adding or subtracting?

- We may have to MULTIPLY before adding or

subtracting!

Pencils down. Watch listen.

- Solve using elimination.
- 3x 5y 11
- 2x 3y 7

Using Multiplication with Elimination

- Write both equations in standard form.
- Look for the easiest way to get one variable to

have opposite coefficients. (Hint Think like

youre finding an LCD). - Multiply EVERY term in the equation by the factor

needed to get the opposites. - Follow the same steps for elimination with

addition or subtraction.

See Example 6, page 257.

- g) -2x 3y 1 h) 8x 4y 0
- -4x y -3 4x 2y 2

What if theres fractions or decimals in the

equations?

- Use the same process weve been using all

semester to get rid of the fractions or decimals.

(multiply by LCD or by powers of 10)

Fractional Coefficients

- Solve using LCD method.

Special Cases

- Dependent equations one equation is a multiple

of the other Solution is INFINITELY MANY

SOLUTIONS. (graph same line) - Note If completely solved, you get a true

statement, like 0 0. - Inconsistent equations have NO SOLUTION (graph

parallel lines) - Note If completely solved, you get a false

statement, like 0 6.

See Examples 8 9, page 258-9.

- j) 3x 2y 1 k) 8x 2y 2
- -3x 2y -1 4x y 2

POSSIBLE SOLUTIONS for a System of Equations

- Answer is a point, (x, y).
- If variables cancel out and you get a true

statement, the solution is all infinitely many

solutions. (Graph would be the same line.) - If variables cancel out and you get a false

statement, there is no solution. (Graph would be

parallel lines.)