D AY 78 – S OLVING S YSTEMS OF E QUATIONS USING S UBSTITUTION
W ORK WITH A PARTNER TO EXPAND YOUR UNDERSTANDING OF SOLVING SYSTEMS WITH SUBSTITUTION . Explain: In the system of equations above, the first equation x is “ ”. 1 y Explain what it means for two expressions to be equal.
A NSWER KEY Explain: In the system of equations above, the first equation x is “ ”. 1 y Explain what it means for two expressions to be equal. They are the same even though they may look different
O BSERVE AND DISCOVER x 1 y x 2 1 7 x y 2 7 x The system of equations is written on the left. The first step of the substitution method is on the right. Explain WHAT was done and WHY it would be OK to do that.
A NSWER K EY Explain WHAT was done and WHY it would be OK to do that. They y was replaced with x + 1. This is “OK” because y = x + 1. y and x + 1 are equivalent.
Using the equation from step 1 of the substitution method (also written below), solve for the variable. (Show your work!) x 2 1 7 x
A NSWER K EY x 2 1 7 x 3 1 7 x 1 1 3 6 x 3 3 2 x
F INISH S OLVING x 1 y y 2 7 x Select either equation from the system and write it here: ___________ Using your answer from the “Begin to Solve” step above, how could you find the value of y? Explain in words. Now actually do the work you just described and find the value of . (Remember to show your work.)
F INISH S OLVING x 1 y Select either equation from the system and write it here: ___________ Using your answer from the “Begin to Solve” step above, how could you find the value of y? Explain in words. I substituted the 2 in the place of the x and then calculated the result Now actually do the work you just described and find the value of . (Remember to show your work.) 2 1 y 3 y
F INISH S OLVING Let’s see if it would make a difference if we had chosen the other equation from the system to find . Write down the equation you didn’t pick … follow the same process … and find the value of again. Was your answer the same?
F INISH S OLVING Let’s see if it would make a difference if we had chosen the other equation from the system to find . Write down the equation you didn’t pick … follow the same process … and find the value of again. y 2 ( 2 ) 7 y 4 7 3 y Was your answer the same? YES
State the values you found for x and y as an ordered pair (x,y): _________________________ Substitute those values for x and y into both equations and simplify. x y 1 2 7 y x
State the values you found for x and y as an ordered pair (x,y): _________________________ Substitute those values for x and y into both equations and simplify. x y 1 2 7 y x 3 2 1 2 ( 2 ) 3 7 3 3 7 7
If the result for both equations is a true statement (examples: 5=5 … 24=24 … -2=- 2 … etc.), that means our ordered pair IS a solution to the system because it satisfies (makes true) both equations.
L ET ’ S GRAPH THE SYSTEM OF EQUATIONS TO CONFIRM OUR SOLUTION FROM A GRAPHICAL PERSPECTIVE . x 1 y Slope= ____ y -intercept= ____ y 2 7 x Solve for y to get in slope- intercept form Slope= ____ y -intercept= ____ Graph the 2 equations on the graph above. State the intersection point: _______ How does this compare to the solution you got using substitution?
L ET ’ S GRAPH THE SYSTEM OF EQUATIONS TO CONFIRM OUR SOLUTION FROM A GRAPHICAL PERSPECTIVE . x 1 y Slope= 1 y -intercept= 1 y 2 7 x Solve for y to get in slope- intercept form Slope= -2 y -intercept= 7 Graph the 2 equations on the graph above. State the intersection point: (2,3) How does this compare to the solution you got using substitution?
S UBSTITUTION Explain each step in solving this system of equation using substitution. ORIGINAL SYSTEM OF EQUATIONS 4 2 y x 3 2 26 y x 3 ( 4 2 ) 2 26 x x Explain in words what’s been done in each step.
S UBSTITUTION Explain each step in solving this system of equation using substitution. ORIGINAL SYSTEM OF EQUATIONS 4 2 y x 3 2 26 y x 3 ( 4 2 ) 2 26 x x Explain in words what’s been done in each step. Substitute 4x + 2 in place of the y
S OLVE THE EQUATION FOR X 12 6 2 26 x x
12 6 2 26 Distribute x x 10 6 26 x 6 6 Solve the 10 20 x equation for x 10 10 2 x
S UBSTITUTE THE X INTO EITHER OF THE ORIGINAL EQUATIONS TO SOLVE FOR Y x x 4 3 2 26 2 y y 3 2 ( 2 ) 26 y 4 ( 2 ) 2 y 3 4 26 y 10 y 4 4 3 30 y 3 3 10 y
N OW YOUR TURN … Solve these systems of equations using substitution x 3 ( 4 ) 2 28 3 2 28 1) x y x x 12 2 28 x 4 y x 14 28 x 4 ( 2 ) 14 14 y 2 8 x y
N OW YOUR TURN … Solve these systems of equations using substitution 2) 3 ( 15 ) 2 5 3 2 5 y y x y 45 3 2 5 y y 15 x y y 45 5 5 15 8 45 45 x y 5 40 7 x 5 5 8 y
W ORD P ROBLEM Josh cannot find a publisher for a book he wrote, so he decides to publish the book himself. He calculates that his fixed costs will be $2665 and that the unit cost of each book will be an additional $1.70. He intends to sell the book for $4.95. How many copies must he sell to break even? ("Break even" means that income equals expenses.)
A NSWER K EY Let x represent the number of copies Josh must sell to break even. Let y represent the break-even amount. Write two equations, one representing expenses and one representing income. Expenses equation 1 . 7 2665 y x Income equation 4 . 95 y x Substitution 4.95 x for y in the ( 4 . 95 ) 1 . 7 2665 x x expenses equation. 3 . 25 2665 Subtract 1.7x from both sides. x 820 Divide both sides by 3.25. x
S OLVE BY THE SUBSTITUTION METHOD : Since the x in equation 1 has a coefficient of -1, it will probably be simpler to solve this equations for x Equation 1 2 4 x y Equation 2 5 3 1 x y
A NSWER K EY 1 2 4 x y 4 2 Multiply each side by (-1) x y 2 4 x y 2 y Replace x with (2y-4) 5 3 1 x 5 ( 2 4 ) 3 1 Solve for y y y 10 20 3 1 y y 7 21 y 3 3 y
3 Replace y with 3 in 3 y Equation 1 1 2 4 x y x 2 ( 3 ) 4 4 2 x {( 2 , 3 )} Solution set:
S OLVE THE SYSTEM BY SUBSTITUTION Equation 1 4 3 4 x y Equation 2 3 2 x y
The variable easiest to isolate is y in Equation 2 Equation 2 3 2 x y Solve for y. 3 2 y x Substitute (-3x-2) for y 4 3 ( 3 2 ) 4 x x in Equation 1. Use the distributive 4 9 6 4 x x Property. Combine like terms. 5 6 4 x Add 6 to both sides. 5 6 6 4 6 x 5 10 Simplify. x Divide both sides 2 x by -5
Having found x, you can now substitute in Equation 2. 3 ( 2 ) 2 Substitute -2 for x y 6 2 Simplify. y 4 Solve for y. y
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