Warm upWarm up

Solve for ySolve for y1.1. 3x + 2y = 53x + 2y = 52.2. -4x – 2y = 8-4x – 2y = 83.3. -6x + 3y = -15-6x + 3y = -15Write an equation in slope intercept formWrite an equation in slope intercept form4.4. m = 4 and y int (0, 3)m = 4 and y int (0, 3)5.5. m = -3/2 and y int (2, 4)m = -3/2 and y int (2, 4)

answersanswers

1.1. y = -3/2x + 5/2y = -3/2x + 5/22.2. y = -2x – 4y = -2x – 43.3. y = 2x – 5y = 2x – 54.4. y = 4x + 3y = 4x + 35.5. y = -3/2x + 7y = -3/2x + 7

Key wordsKey wordsSystem of EquationsSystem of Equations

Two or more equationsTwo or more equations

System SolutionSystem SolutionA point (x, y) that satisfies both A point (x, y) that satisfies both equationsequations

Part 1: Finding solutions to Part 1: Finding solutions to Systems of EquationsSystems of Equations

There are ____ types of solutions There are ____ types of solutions for systems of equations.for systems of equations.

3

One Solution No Solution Infinite Solutions

Types of SolutionsTypes of SolutionsOne Solution: If there is one solution, then the lines

are intersecting. They may or may not be perpendicular. The slopes will be opposite reciprocals if perpendicular.

No Solution: If there is no solution, then the lines are parallel. The slopes will be the same, but “b” will be different.

Infinite Solutions:

If there are infinite solutions, then the lines are coinciding. Both “m” and “b” will be the same.

Solving Systems of EquationsSolving Systems of EquationsThere are 3 ways to solve a system of equations.

Graphing: Graph the lines. Where the lines intersect is the solution.

Substitution: Solve one of the equations for one of the variables and substitute.

Elimination: Set up the equations and combine them to eliminate one of the variables.

Steps for Using EliminationSteps for Using Elimination1)1) Write both equations in standard form Write both equations in standard form

(Ax + By = C) so that variables and = line up(Ax + By = C) so that variables and = line up2)2) Multiply one or both equations by a number to Multiply one or both equations by a number to

make opposite coefficients for one of the make opposite coefficients for one of the variables.variables.

3)3) Add equations together (one variable should Add equations together (one variable should cancel out)cancel out)

4)4) Solve for remaining variable.Solve for remaining variable.5)5) Substitute the solution back in to find other Substitute the solution back in to find other

variable.variable.

Example 1:Example 1: 5x + y = 125x + y = 12 3x – y = 43x – y = 4 8x = 168x = 16 8 88 8 x = 2x = 2

5(2) + y = 125(2) + y = 1210 + y = 1210 + y = 12y = 2y = 2

The solution is: (2, 2)The solution is: (2, 2)

Step 1: Put both equations in standard form.

Step 2: Check for opposite coefficients.

Step 3: Add equations together

Step 4: Solve for x

Step 5: Substitute 2 in for x to solve for y (in either equation)

Already Doney and –y are already opposites

Example 2Example 2 3x + 4y = 93x + 4y = 9 -x – 4y = 7-x – 4y = 7

Answer: (8, -15/4)Answer: (8, -15/4)

Example 3Example 3 3x + 5y = 103x + 5y = 10 3x + y = 23x + y = 2

3x + 5y = 103x + 5y = 10 -1(3x + y) = -1(2-1(3x + y) = -1(2))

4y = 84y = 8 y = 2y = 2Now plug (2) in for y.Now plug (2) in for y.3x + 2 = 23x + 2 = 2X = 0 X = 0 Solution is : (0,2)Solution is : (0,2)

When you add these neither variable drops out

SO….

We need to change 1 or both equations by multiplying the equation by a number that will create opposite coefficients.

When we need to create opposite When we need to create opposite coefficientscoefficients

3x + 5y = 103x + 5y = 10 -3x – y = -2-3x – y = -2

Multiply the bottom equation by negative one to eliminate the x

4) 2x + 3y = 64) 2x + 3y = 6 5x – 4y = -85x – 4y = -8

4(2x + 3y) = 6(4) 8x + 12y = 244(2x + 3y) = 6(4) 8x + 12y = 243(5x – 4y) = -8(3)3(5x – 4y) = -8(3) 15x - 12y = -2415x - 12y = -24

23 x = 023 x = 0 x = 0x = 0

Now plug (0) in for x into any of the 4 equations.Now plug (0) in for x into any of the 4 equations.2(0) + 3y = 62(0) + 3y = 63y = 63y = 6y = 2y = 2Solution is: (0, 2)Solution is: (0, 2)

We will need to change both equations. We will have the y value drop out.

Solving by SubstitutionSolving by Substitution1.1. Solve one equation for either Solve one equation for either x or yx or y2.2. SubstituteSubstitute the expression into the other the expression into the other

equationequation3.3. Solve for the Solve for the variablevariable4.4. SubstituteSubstitute the value back in and solve the value back in and solve5.5. Check your answerCheck your answer, is it a solution for both , is it a solution for both

equations?equations?

RememberRemember that a point consists of an “x” value and a that a point consists of an “x” value and a “y” value. You have to find both to find the solution.“y” value. You have to find both to find the solution.

Step 1Step 1Solve one equation for x or ySolve one equation for x or y

y = x + 1y = x + 1 y = -2x - 2y = -2x - 2

Already done!

Step 2Step 2

Substitute that expression into the other Substitute that expression into the other equationequationyy = = x + 1x + 1

yy = -2x - 2 = -2x - 2

x + 1x + 1 = -2x - 2 = -2x - 2

Step 3Step 3

Solve for the other variableSolve for the other variablex + 1x + 1 = -2x – 2 = -2x – 2

+2x +2x+2x +2x3x + 1 = -23x + 1 = -2 - 1 - 1- 1 - 1 3x = -33x = -3 3 33 3 x = -1x = -1

Step 4Step 4

Substitute the value back in and solveSubstitute the value back in and solve

y = -1+ 1y = 0

Is (-1, 0) a solution? Check to find out.

0 = -1 + 10= -2(-1) – 2

Solution (-1, 0)

Try this oneTry this one

Ex. Ex. y = x + 4y = x + 4y = 3x + 10y = 3x + 10

Solution (-3, 1)

Substitution & the distributive propertySubstitution & the distributive property

To use substitution you must have an To use substitution you must have an equation that has been solved for one of equation that has been solved for one of the variables.the variables.

Ex. 3x – 2Ex. 3x – 2yy = 1 = 1

y= 3 + 1 y= 3 + 1 y = 4 y = 4 Solution: (3, 4)

3x-2(x+1) =13x –2x -2 =1 x -2 = 1 x = 3

y=x+1

Your Turn:Your Turn:

Solve the following systems of equations.Solve the following systems of equations.4. y = x +14. y = x +1 y = 2x – 1y = 2x – 1

5. y = 2x5. y = 2x 7x –y = 157x –y = 15

(2, 3)

(3, 6)

You try these: You try these: Tell if lines are parallel, perpendicular, intersecting Tell if lines are parallel, perpendicular, intersecting

but not perpendicular, or coinciding if one but not perpendicular, or coinciding if one solution, infinite solutions, or many solutions. solution, infinite solutions, or many solutions.

1. 3x + y = 21. 3x + y = 2 2. -4x + 3y = 02. -4x + 3y = 0 3x + y = 63x + y = 6 4x + y = 8 4x + y = 8

3. 5x -10y = 203. 5x -10y = 20 4. y = -1/4 x - 54. y = -1/4 x - 5 -x + 2y = -4-x + 2y = -4 -4x + y = 12 -4x + y = 12

parallel lines, no solution intersecting lines, one solution

Coinciding lines, Infinite solution

perpendicular lines,

one solution

Summary: Draw and Fill in the Summary: Draw and Fill in the table below in your notes.table below in your notes.

Parallel Parallel LinesLines

IntersectingIntersectingNot Not ┴┴

IntersectingIntersectingPerpendicularPerpendicular

Coinciding Coinciding LinesLines

GraphGraph

Slope & bSlope & b

Ex. of what Ex. of what system looks system looks likelikeNumber of Number of SolutionsSolutions

PracticePractice

Classwork: CW #7Classwork: CW #7

Homework: WS #7Homework: WS #7