Review

Ex: Check whether the ordered pairs are solns. of the system. x-3y= -5 -2x+3y=10 A.(1,4) 1-3(4)= = = -5 *doesn’t work in the 1 st eqn, no need to check the 2 nd. Not a solution. B.(-5,0) -5-3(0)= = -5 -2(-5)+3(0)=1010=10Solution

Solving a System Graphically 1.Graph each equation on the same coordinate plane. (USE GRAPH PAPER!!!) 2.If the lines intersect: The point (ordered pair) where the lines intersect is the solution. 3.If the lines do not intersect: a.They are the same line – infinitely many solutions (they have every point in common). b.They are parallel lines – no solution (they share no common points).

Ex: Solve the system graphically. 2x-2y= -8 2x+2y=4 (-1,3)

Ex: Solve the system graphically. 2x+4y=12 x+2y=6  1 st eqn: x-int (6,0) y-int (0,3)  2 ND eqn: x-int (6,0) y-int (0,3)  What does this mean? the 2 eqns are for the same line!  ¸ many solutions

Ex: Solve graphically: x-y=5 2x-2y=9  1 st eqn: x-int (5,0) y-int (0,-5)  2 nd eqn: x-int (9/2,0) y-int (0,-9/2)  What do you notice about the lines?  They are parallel! Go ahead, check the slopes!  No solution!

3-2: Solving Systems of Equations using Substitution

Solving Systems of Equations using Substitution Steps: 1. Solve one equation for one variable (y= ; x= ; a=) 2. Substitute the expression from step one into the other equation. 3. Simplify and solve the equation. 4. Substitute back into either original equation to find the value of the other variable. 5. Check the solution in both equations of the system.

Example #1: y = 4x 3x + y = -21 Step 1: Solve one equation for one variable. y = 4x (This equation is already solved for y.) Step 2: Substitute the expression from step one into the other equation. 3x + y = -21 3x + 4x = -21 Step 3: Simplify and solve the equation. 7x = -21 x = -3

y = 4x 3x + y = -21 Step 4: Substitute back into either original equation to find the value of the other variable. 3x + y = -21 3(-3) + y = y = -21 y = -12 Solution to the system is (-3, -12).

y = 4x 3x + y = -21 Step 5: Check the solution in both equations. y = 4x -12 = 4(-3) -12 = -12 3x + y = -21 3(-3) + (-12) = (-12) = = -21 Solution to the system is (-3,-12).

Example #2: x + y = 10 5x – y = 2 Step 1: Solve one equation for one variable. x + y = 10 y = -x +10 Step 2: Substitute the expression from step one into the other equation. 5x - y = 2 5x -(-x +10) = 2

x + y = 10 5x – y = 2 5x -(-x + 10) = 2 5x + x -10 = 2 6x -10 = 2 6x = 12 x = 2 Step 3: Simplify and solve the equation.

x + y = 10 5x – y = 2 Step 4: Substitute back into either original equation to find the value of the other variable. x + y = y = 10 y = 8 Solution to the system is (2,8).

x + y = 10 5x – y = 2 Step 5: Check the solution in both equations. x + y = =10 10 =10 5x – y = 2 5(2) - (8) = 2 10 – 8 = 2 2 = 2 Solution to the system is (2, 8).

Solve by substitution: 1. 2.

3-2: Solving Systems of Equations using Elimination Steps: 1. Place both equations in Standard Form, Ax + By = C. 2. Determine which variable to eliminate with Addition or Subtraction. 3. Solve for the variable left. 4. Go back and use the found variable in step 3 to find second variable. 5. Check the solution in both equations of the system.

EXAMPLE #1: STEP 2:Use subtraction to eliminate 5x. 5x + 3y =11 5x + 3y = 11 -(5x - 2y =1) -5x + 2y = -1 5x + 3y = 11 5x = 2y + 1 Note: the (-) is distributed. STEP 3:Solve for the variable. 5x + 3y =11 -5x + 2y = -1 5y =10 y = 2 STEP1: Write both equations in Ax + By = C form. 5x + 3y =1 5x - 2y =1

STEP 4: Solve for the other variable by substituting into either equation. 5x + 3y =11 5x + 3(2) =11 5x + 6 =11 5x = 5 x = 1 5x + 3y = 11 5x = 2y + 1 The solution to the system is (1,2).

5x + 3y= 11 5x = 2y + 1 Step 5:Check the solution in both equations. 5x + 3y = 11 5(1) + 3(2) = =11 11=11 5x = 2y + 1 5(1) = 2(2) = =5 The solution to the system is (1,2).

Solving Systems of Equations using Elimination Steps: 1. Place both equations in Standard Form, Ax + By = C. 2. Determine which variable to eliminate with Addition or Subtraction. 3. Solve for the remaining variable. 4. Go back and use the variable found in step 3 to find the second variable. 5. Check the solution in both equations of the system.

Example #2: x + y = 10 5x – y = 2 Step 1: The equations are already in standard form:x + y = 10 5x – y = 2 Step 2: Adding the equations will eliminate y. x + y = 10 +(5x – y = 2) +5x – y = +2 Step 3:Solve for the variable. x + y = 10 +5x – y = +2 6x = 12 x = 2

x + y = 10 5x – y = 2 Step 4: Solve for the other variable by substituting into either equation. x + y = y = 10 y = 8 Solution to the system is (2,8).

x + y = 10 5x – y = 2 x + y = =10 10=10 5x – y =2 5(2) - (8) =2 10 – 8 =2 2=2 Step 5: Check the solution in both equations. Solution to the system is (2,8).

NOW solve these using elimination: NOW solve these using elimination: x + 4y =1 x - 4y =5 2x – y =6 x + y = 3

Using Elimination to Solve a Word Problem: Two angles are supplementary. The measure of one angle is 10 degrees more than three times the other. Find the measure of each angle.

Using Elimination to Solve a Word Problem: Two angles are supplementary. The measure of one angle is 10 more than three times the other. Find the measure of each angle. x = degree measure of angle #1 y = degree measure of angle #2 Therefore x + y = 180

Using Elimination to Solve a Word Problem: Two angles are supplementary. The measure of one angle is 10 more than three times the other. Find the measure of each angle. x + y = 180 x =10 + 3y

Using Elimination to Solve a Word Problem: Solve x + y = 180 x =10 + 3y x + y = 180 -(x - 3y = 10) 4y =170 y = 42.5 x = 180 x = x = (137.5, 42.5)

Using Elimination to Solve a Word Problem: The sum of two numbers is 70 and their difference is 24. Find the two numbers.

Using Elimination to Solve a Word problem: The sum of two numbers is 70 and their difference is 24. Find the two numbers. x = first number y = second number Therefore, x + y = 70

Using Elimination to Solve a Word Problem: The sum of two numbers is 70 and their difference is 24. Find the two numbers. x + y = 70 x – y = 24

Using Elimination to Solve a Word Problem: x + y =70 x - y = 24 2x = 94 x = y = 70 y = 70 – 47 y = 23 (47, 23)

Now you Try to Solve These Problems Using Elimination. Solve 1.Find two numbers whose sum is 18 and whose difference is The sum of two numbers is 128 and their difference is 114. Find the numbers.

MATRIX: A rectangular arrangement of numbers in rows and columns. The ORDER of a matrix is the number of the rows and columns. The ENTRIES are the numbers in the matrix. rows columns This order of this matrix is a 2 x 3.

3 x 3 3 x 5 2 x 2 4 x 1 1 x 4 (or square matrix) (Also called a row matrix) (or square matrix) (Also called a column matrix)

To add two matrices, they must have the same order. To add, you simply add corresponding entries.

= =

To subtract two matrices, they must have the same order. You simply subtract corresponding entries.

= (-1) (-4) =

In matrix algebra, a real number is often called a SCALAR. To multiply a matrix by a scalar, you multiply each entry in the matrix by that scalar.

(-3) -5 -2(6)-2(-5) -2(3)

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