the difference between two numbers is 16.five times the smaller is the same as 8 less than twice the larger.find the numbers
b is big, s is small
b-s = 16
so
s = (b-16)
and also
5 s = 2 b -8
so
5(b-16) = 2 b - 8
5 b - 80 = 2 b - 8
3 b = 72
b = 24
then go back and find s
s = 24-16 = 8
What two numbers should be placed in the blanks below so that each pair of consecutive numbers has the same difference?
13, ____, ____, 49
To find the two numbers, let's call them "x" and "y". We can set up two equations based on the information given:
1) "The difference between two numbers is 16": x - y = 16
2) "Five times the smaller is the same as 8 less than twice the larger": 5x = 2y - 8
We now have a system of two equations with two unknowns, x and y. We can solve this system using either substitution or elimination method.
Let's solve it using the elimination method:
Step 1: Multiply both sides of Equation 1 by 2 to make the coefficients of "y" in both equations the same:
2(x - y) = 2(16)
2x - 2y = 32
Step 2: Now we have two equations:
2x - 2y = 32 (Equation 3)
5x = 2y - 8 (Equation 2)
Step 3: Add Equation 2 and Equation 3 together:
2x - 2y + 5x = 32 + (2y - 8)
7x = 2y + 2y + 32 - 8
7x = 4y + 24
Step 4: Simplify Equation 4:
7x - 4y = 24 (Equation 4)
Step 5: Now we have two equations:
7x - 4y = 24 (Equation 4)
5x = 2y - 8 (Equation 2)
Step 6: Solve the system by eliminating one variable. Let's eliminate "y" by multiplying Equation 4 by 2 and Equation 2 by 4:
14x - 8y = 48 (Equation 5)
20x = 8y - 32 (Equation 6)
Step 7: Add Equation 5 and Equation 6 together:
14x - 8y + 20x = 48 + (8y - 32)
34x = 8y + 8y + 48 - 32
34x = 16y + 16
Step 8: Simplify Equation 7:
34x - 16y = 16 (Equation 7)
Step 9: Now we have two equations:
34x - 16y = 16 (Equation 7)
5x = 2y - 8 (Equation 2)
Step 10: Solve the system by eliminating one variable. Let's eliminate "x" by multiplying Equation 7 by 5 and Equation 2 by 34:
170x - 80y = 80 (Equation 8)
170x = 68y - 272 (Equation 9)
Step 11: Add Equation 8 and Equation 9 together:
170x - 80y + 170x = 80 + (68y - 272)
340x = 80 + 272 + 68y - 80y
340x = 192 + (-12y)
340x + 12y = 192 (Equation 10)
Step 12: Now we have two equations:
340x + 12y = 192 (Equation 10)
5x = 2y - 8 (Equation 2)
Step 13: Solve the system by eliminating one variable. Let's eliminate "x" by multiplying Equation 10 by 5 and Equation 2 by 340:
1700x + 60y = 960 (Equation 11)
1700x = 680y - 2720 (Equation 12)
Step 14: Subtract Equation 12 from Equation 11:
1700x + 60y - 1700x = 960 - (680y - 2720)
60y - 680y = 960 - 2720
-620y = -1760
Step 15: Divide both sides of Equation 13 by -620 to solve for y:
y = (-1760) / (-620)
y = 2.8387 (rounded to 4 decimal places)
Step 16: Substitute the value of y back into Equation 2 to solve for x:
5x = 2(2.8387) - 8
5x = 5.6774 - 8
5x = -2.3226
Step 17: Divide both sides of Equation 15 by 5 to solve for x:
x = (-2.3226) / 5
x = -0.4645 (rounded to 4 decimal places)
Therefore, the two numbers are approximately x = -0.4645 and y = 2.8387.