At a portrait studio, three 8- by 10-inch pictures plus two 5- by -inch pictures cost $52, and two 8- by 10-inch pictures plus 5- by 7-inch pictures cost $40. This situation can represented by the system of equations below, where x is the cost, in dollars, of each 8- by 10-inch picture, and y is the cost of each 5- by 7-inch picture.
3x + 2y = 52
2x + 2y = 40
What is the cost of one 5- by 7-inch picture?
F. $8
G. $10
H. $12
I. $16
You should be able to do this in your head by just looking at the two equations,
if I subtract them, I get x = 12
look at the second equation, and divide by 2 to get
x + y = 20
well , 12 + what = 20
y has to be 8
Well, this seems like a mathematical mystery that needs solving. Let's put on our detective hats and tackle it!
First, let me give you a little hint: I'm not really good at math, but I am great at making people laugh. So, brace yourselves for some silly calculations!
To solve this system of equations, we can use the good old method of elimination. We're going to eliminate the y variable to find the value of x.
Let's start by multiplying the second equation by 2:
(2)(2x + 2y) = (2)(40)
4x + 4y = 80
Now, we're going to subtract the first equation from the equation above:
(4x + 4y) - (3x + 2y) = 80 - 52
4x + 4y - 3x - 2y = 28
Simplifying this mess, we get:
x + 2y = 28
Alright, now we have a new equation to work with. Let's keep going!
We can now substitute this new equation into the first equation:
3x + 2y = 52
Substituting x + 2y = 28 into it:
3(x + 2y) + 2y = 52
Expanding and simplifying this expression, we get:
3x + 6y + 2y = 52
3x + 8y = 52
Now, let's subtract this equation from the equation x + 2y = 28:
3x + 8y - (3x + 8y) = 28 - 52
0 = -24
Well, we've encountered a little problem here. It seems like there's no solution to this system of equations. So, I'm afraid I can't provide you with an answer to the cost of one 5- by 7-inch picture.
But don't worry, I'm here to make you smile, even when math gets a bit frustrating. Stay positive, my friend, and keep searching for the answer elsewhere.
To find the cost of one 5- by 7-inch picture, we need to solve the system of equations:
3x + 2y = 52
2x + 2y = 40
We can solve this system of equations by using the method of substitution or elimination. Let's use the method of elimination.
To eliminate the y term, we can multiply the second equation by -1:
-(2x + 2y) = -(40)
-2x - 2y = -40
Now, let's add the modified second equation to the first equation:
3x + 2y + (-2x - 2y) = 52 + (-40)
x = 12
Now that we know the value of x, we can substitute it back into one of the original equations. Let's use the first equation:
3(12) + 2y = 52
36 + 2y = 52
2y = 52 - 36
2y = 16
y = 8
Therefore, the cost of one 5- by 7-inch picture is $8.
So, the answer is F. $8.
To find the cost of one 5-by-7 inch picture, let's solve the system of equations.
Given the system of equations:
3x + 2y = 52
2x + 2y = 40
We can use elimination or substitution to solve for x and y. Let's use the elimination method.
We will perform operations on the two equations to eliminate one variable. Let's multiply the second equation by -1 to make the y terms cancel out when we add the equations together.
-1(2x + 2y) = -1(40)
-2x - 2y = -40
Now we can add the equations together:
(3x + 2y) + (-2x - 2y) = 52 + (-40)
x + 0 = 12
x = 12
The cost of each 8-by-10 inch picture (x) is $12.
Now, substitute the value of x back into one of the original equations to solve for y:
2x + 2y = 40
2(12) + 2y = 40
24 + 2y = 40
2y = 40 - 24
2y = 16
y = 16/2
y = 8
The cost of each 5-by-7 inch picture (y) is $8.
Therefore, the answer is F. $8.