Number problems of two unknowns?

Question

I'm having a rough time understanding how these problems work. If there is anyone out there who could translate and solve a few for me I would greatly appreciate it. I'm not looking for answers (I already have the answers), I just want to know how the problems work. Thanks!

Here are a few problems:


1) The larger of two numbers is equal to twice the smaller number
increased by 5. The smaller number equals the larger number decreased by 16. What are the numbers?

2) The smaller of two numbers is equal to 2 less than half the larger number. If the larger number, increased by 4, is equal to six times the smaller, what are the two numbers?

3) Three times the smaller of two numbers is equal to twice the larger. If the larger number plus 2 equals twice the smaller number less 4, what are the numbers?

Answer 1

for the first one:
let x=larger number
let y=smaller number

x=2y+5 describes the first condition
y=x-16 describes the second condition

substitute the second equation into the first one:
x=2(x-16)+5
x=2x-32+5
32-5=x
x=27
so, y=x-16=11

for the second one:
x=larger number
y=smaller number
y=(x/2)-2
x+4=6y
substitute the first equation into the second one to get:
x+4=6((x/2)-2)
x+4=(6x/2)-12
x+4=3x-12
16=2x
x=8
so, 8+4=6y
12=6y
y=2

for the 3rd one:
x=larger
y=smaller
3y=2x (rearrange this one to get x=3y/2)
x+2=2y-4

plug the first equation into the second one to get:
(3/2y)+2=2y-4
6=1/2y
y=12
3(12)=2x
36=2x
x=18

always go back and plug the numbers into the original word problem to make sure they check out.

Answer 2

The general rule to solving this type of problem is to understand how to express in algebraic terms the relationships that are given in words. When you see "is equal to," you replace that with an "=" sign.

>1) The larger of two numbers is equal to twice the smaller number
increased by 5. The smaller number equals the larger number decreased by 16. What are the numbers?

a) There are two numbers, but we don't know what they are. Call the first one "m" and the second one "n."
b) One of the numbers is equal to twice the other, increased by 5: m = 2 *n + 5
c) The second number is 16 less than the first number: n = m-16

Now you have 2 equations and two unknowns, and you can solve that
m = 2*n + 5
m-16 = n

>2) The smaller of two numbers is equal to 2 less than half the larger number. If the larger number, increased by 4, is equal to six times the smaller, what are the two numbers?

a) p = (q/2) - 2
p+4 = q * 6

You do the third on your own.

Answer 3

Probably the best insight is to consider each of the equations that you need to write to solve such problems. For example, the first problem. Let x = the smaller number and y = the larger. Then (from the problem)
y = 2x + 5 and
x = y - 16 which is the same as y = x + 16
Now if you graph the straight lines described by these two equations, you will find the the two lines intersect at the coordinates (11,27) or x = 11 and y = 27. This is also the reason that two 'simultaneous' equations may have no solution (in the case that the 2 lines described are parallel) or an infinite number of solutions (in the case that the two lines lay directly on top of each other)

It is only where the two lines intersect, that 'common' values for x and y exist that satisfy both equations 'simultaneously'.

Later on you will discover simultaneous equations in three variables. These equations describe 'planes' and, when they all intersect in a common point, that is the 'simultaneous' solution of all three equations. But that, as they say, is a story for another day ☺


Doug

Answer 4

let me mention solutions
let larger number be x and smaller y for all problems
1)
The larger of two numbers is equal to twice the smaller number
increased by 5.
based on above condition
x = 2y+5 (twice the smaller number + 5= 2y + 5)

The smaller number equals the larger number decreased by 16. What are the numbers?
larger number decreased by 16 is x-16 same as smaller number

so x- 16 = y

2) The smaller of two numbers is equal to 2 less than half the larger number. If the larger number, increased by 4, is equal to six times the smaller, what are the two numbers?
2 less than 1/2 the larger number = x/2-2 = y
larger number increased by 4 = x+ 4
6 times smaller number = 6y = x+4


you can work out the 3rd

Answer 5

1) let the smaller no. be x and the larger one be y
Given y = 2x + 5 or that is y-2x = 5
y-x = 16

Subtracting the two equations, we get
-x = - 11 or that is x = 11
Now substitue the value of x in the second equation, we get y-11 = 16 or y = 16+11 = 27

Hence the smaller no. is 11 and the larger no. is 27


2) Let the smaller no. be x and the larger no. be y
Given x = y/2 - 2 or that is x= y-4/2 or that is 2x=y-4 or that
is 2x-y=-4
y+4 = 6x or that is y-6x = -4
Now adding the two equations we get -4x= -8 or x = 2
substituting in the first equation for x we get 4 = y-4 or y = 4+4=8

Hence the smaller no. is 2 and the larger one is 8

Please solve the third one in the same manner

Answer 6

First, set up the word problems as algabraic expressions, using X and Y to represent the variables.
In problem 1, using x for the larger number and y for the smaller number, you would come up with, x = 2y + 5 and y = x - 16.

Second, solve one of the equations so that we get an equivalance for one of the variables. In Problem 1, we already have x = 2y + 5, so this one has been done for us.

Third, replace x with its equal value of y. In problem 1, we see that x = 2y + 5. so in the second equation, substitute 2y + 5 whereever you see x. This results in y = (2y + 5) - 16.

Finally, solve the new equation for y.
y = 2y - 11
y + 11 = 2y
11 = y and therefore, x = 27

Good luck.

Answer 7

Let "x" the larger number, "y" the smaller.
1) you have the system x = 2y + 5
y = x - 16
2) y = 2 - x/2
x + 4 = 6y
3) 3y = 2x
x + 2 = 2y - 4
Now solve this systems and you get the numbers.

Answer 8

(1)
x = 2(y+5)
y = x - 16
=> x = 2(x-16 + 5)
=> x = 2x - 22 => x = 22 and y = 6

(2)
y = 2 -1/2 x
4x = 6y => x = 3/2 y
=> y = 2 - 3/4y => 7/4y = 2 => y=8/7 and x = 12/7

(3)
3y = 2x => x=3/2y (a)
x + 2 = 2(y - 4)=> x = 2y - 8 - 2 = 2y -10 (b)
(a) = (b)
3/2 y = 2y -10
1/2 y = 10 => y = 20 and x = 30

Answer 9

1) x > y
so x = 2y + 5
y = x - 16