Write a linear system of equations that can be used to solve these problems. Then, solve to get your final answer. Please help me work through this.
4. The difference of two numbers is 3. Their sum is 13. Find the numbers.
5. Matt and Michelle are selling fruit. Customers can buy small boxes of oranges and large boxes of oranges. Matt sold 3 small boxes of oranges and 14 large box oranges for a total of $203. Michelle sold 11 small boxes of oranges and 11 large boxes of oranges for a total of $220. Find the cost of each small box of oranges and each large box of oranges.
Very similar to the ones I just did for you.
#4 is very very easy.
Show me your steps for #4
To form a linear system of equations, we need to assign variables to the unknown quantities in the problems.
4. Let's say the two numbers are x and y.
The difference of two numbers is 3:
x - y = 3
Their sum is 13:
x + y = 13
To solve this system of equations, we can use the method of substitution or elimination. Let's solve using the elimination method:
From the given equations:
x + y = 13 (Equation 1)
x - y = 3 (Equation 2)
When we add Equation 1 and Equation 2, the y variable will be eliminated:
(x + y) + (x - y) = 13 + 3
2x = 16
Dividing both sides by 2, we get:
x = 8
Now, let's substitute the value of x back into Equation 1 to solve for y:
8 + y = 13
y = 13 - 8
y = 5
Therefore, the two numbers are 8 and 5.
5. Let's say the cost of a small box of oranges is x dollars, and the cost of a large box of oranges is y dollars.
From the given information:
Matt sold 3 small boxes of oranges and 14 large boxes of oranges for a total of $203:
3x + 14y = 203 (Equation 1)
Michelle sold 11 small boxes of oranges and 11 large boxes of oranges for a total of $220:
11x + 11y = 220 (Equation 2)
To solve this system of equations, we can use the method of substitution or elimination. Let's solve using the elimination method by multiplying Equation 1 by 11 and Equation 2 by 3 to eliminate the x variable:
33x + 154y = 2233 (Equation 3)
33x + 33y = 660 (Equation 4)
When we subtract Equation 4 from Equation 3, the x variable will be eliminated:
(33x + 154y) - (33x + 33y) = 2233 - 660
121y = 1573
Dividing both sides by 121, we get:
y = 13
Now, let's substitute the value of y back into Equation 1 or Equation 2 to solve for x:
3x + 14(13) = 203
3x + 182 = 203
3x = 203 - 182
3x = 21
Dividing both sides by 3, we get:
x = 7
Therefore, the cost of a small box of oranges is $7, and the cost of a large box of oranges is $13.