Note: Enter your answer and show all the steps that you use to solve this problem in the space provided. Use substitution or elimination to find the solution of the following system x+3y=5 x+4y=6
To solve the system of equations using substitution, we can solve one equation for one variable and substitute the expression into the other equation.
From the first equation, we can solve for x in terms of y:
x + 3y = 5
x = 5 - 3y
Substitute this expression for x in the second equation:
(5 - 3y) + 4y = 6
5 - 3y + 4y = 6
5 + y = 6
y = 6 - 5
y = 1
Substitute the value of y back into the expression for x:
x = 5 - 3(1)
x = 5 - 3
x = 2
Therefore, the solution to the system of equations is x = 2 and y = 1.
To find the solution of the given system of equations using the substitution or elimination method, we will start with the substitution method in this case.
Step 1: Solve one equation for one variable in terms of the other variable.
Let's solve the first equation, x + 3y = 5, for x in terms of y.
x = 5 - 3y
Step 2: Substitute the expression found in Step 1 into the other equation.
Substitute x = 5 - 3y into the second equation, x + 4y = 6.
(5 - 3y) + 4y = 6
Step 3: Solve the equation obtained in Step 2 for the remaining variable.
Combine like terms: 5 - 3y + 4y = 6
Simplify: 5 + y = 6
Subtract 5 from both sides: y = 6 - 5
Simplify: y = 1
Step 4: Substitute the value of y found in Step 3 into the expression for x obtained in Step 1.
Substitute y = 1 into x = 5 - 3y.
x = 5 - 3(1)
x = 5 - 3
x = 2
Step 5: Check if the found values of x and y satisfy both equations in the system.
Substitute x = 2 and y = 1 into the original equations:
First equation: 2 + 3(1) = 5
2 + 3 = 5
5 = 5 (True)
Second equation: 2 + 4(1) = 6
2 + 4 = 6
6 = 6 (True)
Since both equations are satisfied, the solution to the system is x = 2 and y = 1.
Final Answer:
The solution to the system x + 3y = 5 and x + 4y = 6 is x = 2 and y = 1.
To solve the system of equations using elimination, you can choose to eliminate either the x or y variable. In this case, let's eliminate the x variable.
Let's multiply the first equation by -1 and then add the two equations together to eliminate x:
Step 1: Multiply the first equation by -1
-1(x + 3y) = -1(5)
- x - 3y = -5
Step 2: Add the two equations together
(-x - 3y) + (x + 4y) = -5 + 6
Simplifying the equation:
-x + x - 3y + 4y = 1
y = 1
Now that you have the value of y, you can substitute it back into one of the original equations to solve for x:
Using the first equation:
x + 3(1) = 5
x + 3 = 5
x = 5 - 3
x = 2
Thus, the solution to the system of equations is x = 2 and y = 1.