the sum of two numbers is 40. the sum of twice the larger and 4 times the smaller is 108. find the numbers

Question

systems of linear equations

applications: distance-rate-time, number problems, amount and cost

 

Answer 1

For this problem, we can see that the information provided can be broken into two separate equations. Since we don't know what the numbers being added are yet, we will have to plug in variables.

 

The sum of two numbers is 40 translates into x + y = 40.

 

The sum of twice the larger and 4 times the smaller is 108 translates into 2x + 4y= 108.

 

Now that we have our two equations, let's solve for x in the first, simpler equation.

 

x + y = 40

x = 40 - y

 

Now that we have a value for x, we can plug that "value" into our second equation and solve for y. 

 

2x + 4y = 108

2(40-y) + 4y = 108 Here we will have to distribute the 2 on the left side of the equation.

80 - 2y + 4y = 108 Next, let's combine the like terms.

80 + 2y = 108 Now subtract 80 from both sides.

2y = 28  Now divide by 2 to solve for y.

y = 14    Now that we have an actual numerical value for y, let's plug this back into our first original equation.

 

x + y = 40

x + 14 = 40

x = 26

 

Now that we have a value for x and a value for y, we can check our work by plugging them into our second original equation and solving.

 

2x + 4y = 108

2(26) + 4(14) = 108

52 + 56 = 108

108 = 108

 

Since both sides of the equation are equal, we proved that our values for x and y are correct.