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Use substitution or elimination to find the solution to the system of equations. Show all work.
2x+4y=10
3x+5y=11
To solve this system of equations using substitution or elimination, we can start by solving one equation for one variable and then substitute it into the other equation. Let's choose to solve the first equation for x.
Step 1: Solve the first equation for x by isolating x.
2x + 4y = 10
Subtract 4y from both sides:
2x = 10 - 4y
Divide both sides by 2:
x = (10 - 4y) / 2
x = 5 - 2y
Now, we have an expression for x in terms of y, which we can substitute into the second equation:
Step 2: Substitute x = 5 - 2y into the second equation.
3x + 5y = 11
Replace x with the expression we found:
3(5 - 2y) + 5y = 11
Distribute the 3:
15 - 6y + 5y = 11
Combine like terms:
15 - y = 11
Subtract 15 from both sides:
-y = -4
Multiply both sides by -1 to solve for y:
y = 4
Now that we have the value for y, we can substitute it back into the expression we found for x:
Step 3: Substitute y = 4 into x = 5 - 2y.
x = 5 - 2(4)
x = 5 - 8
x = -3
Therefore, the solution to the system of equations is x = -3 and y = 4.
To solve this system of equations, we can use either substitution or elimination method.
Method 1: Substitution
Step 1: Solve one equation for one variable in terms of the other variable.
Let's solve the first equation for x:
2x + 4y = 10
2x = 10 - 4y
x = 5 - 2y
Step 2: Substitute this expression for x in the second equation.
3x + 5y = 11
3(5 - 2y) + 5y = 11
15 - 6y + 5y = 11
-y = 11 - 15
-y = -4
y = 4
Step 3: Substitute the value of y back into x = 5 - 2y to find x.
x = 5 - 2(4)
x = 5 - 8
x = -3
The solution to the system of equations is x = -3 and y = 4.
Method 2: Elimination
Step 1: Multiply the first equation by 3 and the second equation by 2 to make the coefficients of x the same.
3(2x + 4y) = 3(10)
2(3x + 5y) = 2(11)
6x + 12y = 30
6x + 10y = 22
Step 2: Subtract the second equation from the first equation to eliminate x.
(6x + 12y) - (6x + 10y) = 30 - 22
2y = 8
y = 4
Step 3: Substitute the value of y back into the first equation to find x.
2x + 4(4) = 10
2x + 16 = 10
2x = 10 - 16
2x = -6
x = -3
The solution to the system of equations is x = -3 and y = 4.
To solve this system of equations using substitution or elimination, let's use the elimination method.
We can start by multiplying the first equation by 3 and the second equation by 2 to make the coefficients of x in both equations equal:
3 * (2x + 4y) = 3 * 10
2 * (3x + 5y) = 2 * 11
Simplifying, we get:
6x + 12y = 30
6x + 10y = 22
Now, we can subtract the second equation from the first equation to eliminate the x terms:
(6x + 12y) - (6x + 10y) = 30 - 22
Simplifying, we get:
6y = 8
Next, we divide both sides of the equation by 6:
6y/6 = 8/6
Simplifying further, we find:
y = 4/3
Now, we can substitute this value of y back into one of the original equations to solve for x. Let's use the first equation:
2x + 4(4/3) = 10
Multiplying, we get:
2x + 16/3 = 10
To eliminate the fraction, we can multiply the entire equation by 3:
3 * (2x + 16/3) = 3 * 10
Simplifying, we get:
6x + 16 = 30
Next, we can isolate the x term by subtracting 16 from both sides of the equation:
6x + 16 - 16 = 30 - 16
Simplifying further, we find:
6x = 14
Finally, we divide both sides of the equation by 6 to solve for x:
6x/6 = 14/6
Simplifying, we get:
x = 7/3
So, the solution to the system of equations is x = 7/3 and y = 4/3.