The sum of two numbers is 70. One number is 10 more than twice the other number. Using Elimination find the numbers.
x+y = 70
y = 2x+10
Now go for it.
Let's assume that the first number is represented by x, and the second number is represented by y.
From the problem statement, we can create two equations:
Equation 1: The sum of two numbers is 70.
x + y = 70
Equation 2: One number is 10 more than twice the other number.
x = 2y + 10
To solve this system of equations using the elimination method, we need to eliminate one variable (either x or y) from the equations by manipulating them.
Step 1: Multiply Equation 1 by -2 to create opposite coefficients for x.
-2(x + y) = -2(70)
-2x - 2y = -140
Step 2: Rewrite Equation 2 in standard form.
x - 2y = 10
Step 3: Add the modified Equations 1 and 2.
(-2x - 2y) + (x - 2y) = -140 + 10
-x - 4y = -130
Now we have a new equation: -x - 4y = -130.
Step 4: Multiply Equation 2 by -1 to create opposite coefficients for x.
(-1)(x - 2y) = -1(10)
-x + 2y = -10
Step 5: Add the newly modified Equation 1 and Equation 2.
(-x - 4y) + (-x + 2y) = -130 + (-10)
-2x - 2y = -140
Now we have a new equation: -2x - 2y = -140.
Step 6: Divide the new Equation by -2 to simplify.
(-2x - 2y) / -2 = -140 / -2
x + y = 70
We can observe that the new equation (x + y = 70) is the same as the original Equation 1. This means that the two equations are equivalent and we can conclude that the original system of equations is dependent.
Therefore, there are infinitely many solutions to this system of equations.
To solve this problem using the method of elimination, we'll need to set up a system of equations based on the given information.
Let's assume the two numbers are x and y.
Based on the first statement, "The sum of two numbers is 70," we can write our first equation as:
x + y = 70 ...(Equation 1)
According to the second statement, "One number is 10 more than twice the other number," we can express the second number (y) as:
y = 2x + 10 ...(Equation 2)
Now, we'll use the method of elimination to solve this system of equations. We want to eliminate one variable so that we can solve for the other.
To eliminate y, we need to eliminate y from our second equation. We can do this by substituting the value of y from Equation 2 into Equation 1.
Substituting (2x + 10) for y in Equation 1, we get:
x + (2x + 10) = 70
Combining like terms, we have:
3x + 10 = 70
Next, we'll isolate x by subtracting 10 from both sides:
3x = 60
Finally, we can solve for x by dividing both sides by 3:
x = 20
Now that we have the value of x, we can substitute it back into Equation 2 to find y:
y = 2x + 10
y = 2(20) + 10
y = 40 + 10
y = 50
Therefore, the two numbers are 20 and 50.