the difference between two numbers is 14. if two times the smaller is added to one half the larger, the result is 52. find the numbers.

Question

Systems of linear equtions and inequalities

Answer 1

We have 2 unknowns so we must have 2 equations to solve this.

Let x = the larger unknown

y = the smaller unknown

Eq. I: x - y = 14

Eq. II: 0.5x + 2y = 52

We can use either the substitution or elimination method to solve this. Let's use the elimination method. Start by multiplying Eq. I by 2 and then adding it to Eq. II and solve for x:

x - y = 14 Eq. I

2x - 2y = 28 Eq. I multiplied by 2

0.5x + 2y = 52 Eq. II

2.5x = 80 summation of adding Eq. I and II

x = 80/2.5 divide both sides by 2.5 to isolate x

x = 32 solve for x

Use this value for x in Eq. I and solve for y:

x - y = 14 Eq. I

32 - y = 14 substitute 32 for x

32 - 14 = y add y to and subtract y from both sides

18 = y solve for y

Therefore, your answers are x = 32 and y = 18..

Answer 2

You have 2 unknowns, so you must have 2 equations to solve this.

Let x = larger unknown

y = smaller unknown

Eq. I: x - y = 14

Eq. II: 2y + (1/2)x = 52 or 2y + 0.5y = 52

You can use either the substitution or elimination method to solve for this. Let's use the elimination method. Start by multiplying Eq. I by 2 and adding this to Eq. II to solve for x. Then take this value for x, put it into either Eq. I or II and solve for y:

x - y = 14 Eq. I

2x - 2y = 28 Eq. I multiplied by 2

0.5x + 2y = 52 Eq. II

2.5x = 80 summation of adding the two equations

x = 80/2.5 divide both sides by 2.5 to isolate x

x = 32 solve for x

Use this value in Eq. I and solve for y:

x - y = 14 Eq. I

32 - y = 14 substitute 32 for x

32 - 14 = y add y to and subtract 14 from both sides

18 = y solve for y

Check both equations for these values of x and y and you will see that they are correct. Therefore, x is 32 and y is 18.