Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Cubic function wikipedia , lookup
Eisenstein's criterion wikipedia , lookup
Signal-flow graph wikipedia , lookup
Quadratic equation wikipedia , lookup
Quartic function wikipedia , lookup
Elementary algebra wikipedia , lookup
History of algebra wikipedia , lookup
System of polynomial equations wikipedia , lookup
March 18, 2014 Pg. 357 #9-14 9.)ππ β πππ = βππ ππ + πππ = ππ STEP 1: Start with original equation given. 3π₯ β 10π¦ = β25 4π₯ + 40π¦ = 20 STEP2: Modify the equations by multiplying by a constant, which are the opposite coefficient for the x variable of the other equation. π(3π₯ β 10π¦ = β25) π(4π₯ + 40π¦ = 20) STEP3: Select ONE of the constants to change sign from original sign. β4(3π₯ β 10π¦ = β25) 3(4π₯ + 40π¦ = 20) STEP4: Distribute the constants to the equations. β12π₯ + 40π¦ = 100 12π₯ + 120π¦ = 60 STEP5: Add the coefficients for the x and y variable and constant vertically. 0π₯ + 160π¦ = 160 160π¦ = 160 STEP6: Solve for the y variable using division. 160π¦ 160 = 160 160 Solution for y variable. π=π STEP7: Re-substitute the y variable into one equation from STEP 1. 3π₯ β 10π¦ = β25 3π₯ β 10(1) = β25 STEP8: Multiply 3π₯ β 10 = β25 STEP 9: Move the constant to right side by adding or subtracting the value of the constant. 3π₯ β 10 = β25 +10 + 10 3π₯ = β15 STEP10: Divide by the coefficient of the variable. 3π₯ β15 = 3 3 π = βπ STEP11: Final Solution (βπ, π) 10.) ππ + πππ = ππ π β ππ = ππ STEP 1: Start with original equation given. 7π₯ + 15π¦ = 32 π₯ β 3π¦ = 20 STEP2: Modify the equations by multiplying by a constant, which are the opposite coefficient for the x variable of the other equation. π(7π₯ + 15π¦ = 32) π(π₯ β 3π¦ = 20) STEP3: Select ONE of the constants to change sign from original sign. 1(7π₯ + 15π¦ = 32) β7(π₯ β 3π¦ = 20) STEP4: Distribute the constants to the equations. 7π₯ + 15π¦ = 32 β7π₯ + 21π¦ = β140 STEP5: Add the coefficients for the x and y variable and constant vertically. 0π₯ + 36π¦ = β108 36π¦ = β108 STEP6: Solve for the y variable using division. 36π¦ β108 = 36 36 Solution for y variable. π = βπ STEP7: Re-substitute the y variable into one equation from STEP 1. π₯ β 3π¦ = 20 π₯ β π(βπ) = 20 STEP8: Multiply π₯ + 9 = 20 STEP 9: Move the constant to right side by adding or subtracting the value of the constant. π₯ + 9 = 20 β9 β 9 π₯ = 11 STEP10: Final Solution (ππ, βπ) π β ππ = ππ βπππ + πππ = βπ STEP 1: Start with original equation given. π₯ β 8π¦ = 18 β16π₯ + 16π¦ = β8 STEP2: Modify the equations by multiplying by a constant, which are the opposite coefficient for the x variable of the other equation. βππ(π₯ β 8π¦ = 18) π(β16π₯ + 16π¦ = β8) 11.) STEP3: Select ONE of the constants to change sign from original sign. ππ(π₯ β 8π¦ = 18) 1(β16π₯ + 16π¦ = β8) STEP4: Distribute the constants to the equations. 16π₯ β 128π¦ = 288 β16π₯ + 16π¦ = β8 STEP5: Add the coefficients for the x and y variable and constant vertically. 0π₯ + β112π¦ = 280 β112π¦ = 280 STEP6: Solve for the y variable using division. β112π¦ 280 = β112 β112 Solution for y variable. βπ π = βπ. ππ ππ π STEP7: Re-substitute the y variable into one equation from STEP 1. π₯ β 8π¦ = 18 π₯ β 8(β2.5) = 18 STEP8: Multiply π₯ + 20 = 18 STEP 9: Move the constant to right side by adding or subtracting the value of the constant. π₯ + 20 = 18 β20 β 20 π₯ = β2 STEP11: Final Solution βπ (βπ, βπ. ππ) ππ (βπ, ) π 12.) πππ + ππ = ππ ππ β ππ = βππ STEP 1: Start with original equation given. 24π₯ + 2π¦ = 52 6π₯ β 3π¦ = β36 STEP2: Modify the equations by multiplying by a constant, which are the opposite coefficient for the x variable of the other equation. π(24π₯ + 2π¦ = 52) ππ(6π₯ β 3π¦ = β36) STEP3: Select ONE of the constants to change sign from original sign. β6(24π₯ + 2π¦ = 52) 24(6π₯ β 3π¦ = β36) STEP4: Distribute the constants to the equations. β144π₯ β πππ¦ = β312 144π₯ β πππ¦ = β864 STEP5: Add the coefficients for the x and y variable and constant vertically. 0π₯ + β84π¦ = β1176 β84π¦ = β1176 STEP6: Solve for the y variable using division. β84π¦ β1176 = β84 β84 Solution for y variable. π = ππ STEP7: Re-substitute the y variable into one equation from STEP 1. 6π₯ β 3π¦ = β36 6π₯ β 3(14) = β36 STEP8: Multiply 6π₯ β 42 = β36 STEP 9: Move the constant to right side by adding or subtracting the value of the constant. 6π₯ β 42 = β36 +42 + 42 6π₯ = 6 STEP10: Divide by the coefficient of the variable. 6π₯ 6 = 6 6 π=π STEP11: Final Solution (π, ππ) 13.) πππ β ππ = ππ βππ + ππ = βπ STEP 1: Start with original equation given. 88π₯ β 5π¦ = 39 β8π₯ + 3π¦ = β1 STEP2: Modify the equations by multiplying by a constant, which are the opposite coefficient for the x variable of the other equation. βπ(88π₯ β 5π¦ = 39) ππ(β8π₯ + 3π¦ = β1) STEP3: Select ONE of the constants to change sign from original sign. π(88π₯ β 5π¦ = 39) 88(β8π₯ + 3π¦ = β1) STEP4: Distribute the constants to the equations. 704π₯ β 40π¦ = 312 β704π₯ + 264π¦ = β88 STEP5: Add the coefficients for the x and y variable and constant vertically. 0π₯ + 224π¦ = 224 224π¦ = 224 STEP6: Solve for the y variable using division. 224π¦ 224 = 224 224 Solution for y variable. π=π STEP7: Re-substitute the y variable into one equation from STEP 1. β8π₯ + 3π¦ = β1 β8π₯ + 3(1) = β1 STEP8: Multiply β8π₯ + 3 = β1 STEP 9: Move the constant to right side by adding or subtracting the value of the constant. β8π₯ + 3 = β1 β3 β 3 -8π₯ = β4 STEP10: Divide by the coefficient of the variable. β8π₯ β4 = β8 β8 π π =. π πΆπΉ π STEP11: Final Solution π ( , π) πΆπΉ (. π, π) π 14.) ππ + ππ = π ππ + π = βπ STEP 1: Start with original equation given. 2π₯ + 4π¦ = 8 5π₯ + π¦ = β7 STEP2: Modify the equations by multiplying by a constant, which are the opposite coefficient for the x variable of the other equation. π(2π₯ + 4π¦ = 8) π(5π₯ + π¦ = β7) STEP3: Select ONE of the constants to change sign from original sign. βπ(2π₯ + 4π¦ = 8) 2(5π₯ + π¦ = β7) STEP4: Distribute the constants to the equations. β10π₯ β 20π¦ = β40 10π₯ + 2π¦ = β14 STEP5: Add the coefficients for the x and y variable and constant vertically. 0π₯ β 18π¦ = β54 β18π¦ = β54 STEP6: Solve for the y variable using division. β18π¦ β54 = β18 β18 Solution for y variable. π=π STEP7: Re-substitute the y variable into one equation from STEP 1. 2π₯ + 4π¦ = 8 2π₯ + 4(3) = 8 STEP8: Multiply 2π₯ + 12 = 8 STEP 9: Move the constant to right side by adding or subtracting the value of the constant. 2π₯ + 12 = 8 β12 β 12 2π₯ = β4 STEP10: Divide by the coefficient of the variable. 2π₯ β4 = 2 2 π = βπ STEP11: Final Solution (βπ, π)