Solving Systems of Linear Equations: 2x - 3y/4 = 3 and 5x = 2y + 7 by Substitution Method
In this article, we will solve a system of linear equations using the substitution method. The system of equations is:
Equation 1: 2x - 3y/4 = 3 Equation 2: 5x = 2y + 7
Step 1: Solving Equation 2 for y
We can solve Equation 2 for y by rearranging the terms:
5x = 2y + 7
Subtracting 7 from both sides gives:
5x - 7 = 2y
Dividing both sides by 2 gives:
(5x - 7)/2 = y
Step 2: Substituting the expression for y into Equation 1
Now, we can substitute the expression for y into Equation 1:
2x - 3((5x - 7)/2)/4 = 3
Step 3: Simplifying the expression
Simplifying the expression, we get:
2x - 3(5x - 7)/8 = 3
Expanding the parentheses and combining like terms, we get:
2x - 15x/8 + 21/8 = 3
Step 4: Solving for x
Multiplying both sides by 8 to eliminate the fractions, we get:
16x - 15x + 21 = 24
Combining like terms, we get:
x + 21 = 24
Subtracting 21 from both sides gives:
x = 3
Step 5: Finding the value of y
Now that we have found the value of x, we can substitute it back into the expression for y:
y = (5x - 7)/2 = (5(3) - 7)/2 = (15 - 7)/2 = 8/2 = 4
Solution
Therefore, the solution to the system of linear equations is x = 3 and y = 4.
Verification
We can verify the solution by plugging the values back into both equations:
Equation 1: 2(3) - 3(4)/4 = 3 = 6 - 3 = 3 (True)
Equation 2: 5(3) = 2(4) + 7 = 15 = 15 (True)
The solution satisfies both equations, confirming that x = 3 and y = 4 is the correct solution.
