substitution

1) The number a is 70% less than the positive number b 70% less means 100% - 70% = 30% 30% is 0.3 a = 0.3b 2) The number c is 80% greater than a 80% less means 100% + 80% = 180% 180% is 1.8 c = 1.8a We want c in terms of b. Substitute equation 1 into equation 2: c = 1.8(0.3b) c = 0.54b

If f(x)=4x²-x+3, then what is f(a)? f(a) = 4a^2 - a + 3

If a is 4 and b is 6, what is the value of this expression? sqrt(a^2+2b-3) Plug in our values for a and b: sqrt(4^2+2(6)-3) sqrt(16 + 12 - 3) sqrt(25) 5

if x=2, and y=3, what is x+y x + y = 2 + 3 x + y = 5

If j = –1 and k = 16, what is k + 16j Plug in our numbers: 16 + 16 * -1 16 - 16 0

A barn contains cows, ducks, and a 3-legged dog named Tripod. There are twice as many cows as ducks in the barn and a total of 313 legs. How many ducks are there in the barn? Let the number of ducks be d. Duck legs = 2 * d = 2d Number of cows = 2d. Cow legs = 4 * 2d = 8d 1 dog Tripod has 3...

Luke and Dan's total debt is $72. If Luke's debt is three times Dan's debt, what is Dan's debt? Let Dan's debt be d. Let Luke's debt be l. We're given two equations: d + l = 72 l = 3d Substitute equation (2) for l into equation (1): d + 3d = 72 Solve for d in the equation d + 3d = 72 Step...

A 100 point test contains a total of 20 questions. The multiple choice questions are worth 3 points each and short response questions are worth 8 points each. Write a system of linear equations that represents this situation Assumptions: Let m be the number of multiple choice questions Let s...

4 adults and 3 children cost $40. Two adults and 6 children cost $38 Givens and Assumptions: Let the number of adults be a Let the number of children be c Cost = Price * Quantity We're given 2 equations: 4a + 3c = 40 2a + 6c = 38 We can solve this system of equations 3 ways Substitution...

larger of 2 numbers is 12 more than the smaller number. if the sum of the 2 numbers is 74 find the 2 numbers Declare Variables for each number: Let l be the larger number Let s be the smaller number We're given two equations: l = s + 12 l + s = 74 Equation (1) already has l solved for...

A man purchased 20 tickets for a total of $225. The tickets cost $15 for adults and $10 for children. What was the cost of each ticket? Declare variables: Let a be the number of adult's tickets Let c be the number of children's tickets Cost = Price * Quantity We're given two equations: a +...

In a class there are 5 more boys than girls. There are 13 students in all. How many boys are there in the class? We start by declaring variables for boys and girls: Let b be the number of boys Let g be the number of girls We're given two equations: b = g + 5 b + g = 13 Substitute equation...

Alberto and Willie each improved their yards by planting daylilies and ivy. They bought their supplies from the same store. Alberto spent $64 on 3 daylilies and 8 pots of ivy. Willie spent $107 on 9 daylilies and 7 pots of ivy. What is the cost of one daylily and the cost of one pot of ivy...

According to the American Bureau of Labor Statistics, you will devote 32 years to sleeping and eating. The number of years sleeping will exceed the number of years eating by 24. Over your lifetime, how many years will you spend on each of these activities? Assumptions: Let years eating be e...

At the end of the 2021 NBA season, the NY Knicks had 10 more wins than losses. This NBA season the NY Knicks played a total of 72 times. Find a solution to this problem and explain. Let w be the number of wins Let l be the number of losses We're given two equations: w = l + 10 l + w = 72 To...

A first number plus twice a second number is 10. Twice the first number plus the second totals 35. Find the numbers. The phrase a number means an arbitrary variable A first number is written as x A second number is written as y Twice a second number means we multiply y by 2: 2y A first number...

the cost of a buffet at a restaurant is different for adults and kids. the bill for 2 adults and 3 kids is $51. the bill for 3 adults and 1 kid is $45. what is the cost per adult and per kid? Let the cost for each adult be a Let the cost for each kid be k We're given two equations: 2a + 3k =...

A straight line has the equation ax + by=23. The points (5,-2) and (1,-5) lie on the line. Find the values of a and b. plug in both points and form 2 equations: 5a - 2b = 23 1x - 5b = 23 We can solve this simultaneous equations any one of three ways: Substitution Method Elimination Method...

2 pens and 1 eraser cost $35 and 3 pens and 4 erasers cost $65. X represents the cost of 1 pen and Y represents the cost of 1 eraser. Write the 2 simultaneous equations and solve. Set up our 2 equations where cost = price * quantity: 2x + y = 35 3x + 4y = 65 We can solve this one of three...

at a bakery the cost of one cupcake and 2 slices of pie is $12.40. the cost of 2 cupcakes and 3 slices of pie costs $20.20. what is the cost of one cupcake? Let the number of cupcakes be c Let the number of pie slices be p Total Cost = Unit cost * quantity So we're given two equations: 1c +...