Solving Systems of Linear Equations: 3x - 5y - 4 = 0 and 9x = 2y + 7 by Substitution Method
In this article, we will learn how to solve a system of linear equations using the substitution method. Specifically, we will solve the following system of equations:
Equations:
- 3x - 5y - 4 = 0 ... (1)
- 9x = 2y + 7 ... (2)
What is the Substitution Method?
The substitution method is a technique used to solve systems of linear equations. It involves solving one equation for one variable and then substituting that expression into the other equation.
Step 1: Solve one of the equations for one variable
We will solve equation (2) for x.
9x = 2y + 7
x = (2y + 7) / 9 ... (3)
Step 2: Substitute the expression into the other equation
We will substitute the expression for x from equation (3) into equation (1).
3((2y + 7) / 9) - 5y - 4 = 0
Step 3: Simplify and solve for y
Multiply both sides by 9 to eliminate the fraction.
3(2y + 7) - 45y - 36 = 0
Expand and simplify.
6y + 21 - 45y - 36 = 0
Combine like terms.
-39y - 15 = 0
Add 15 to both sides.
-39y = 15
Divide both sides by -39.
y = -15 / 39 y = -5 / 13
Step 4: Substitute the value of y back into one of the original equations to solve for x
We will substitute the value of y back into equation (3).
x = (2(-5/13) + 7) / 9
x = (-10/13 + 7) / 9
x = (91/13) / 9
x = 91 / 117 x = 7 / 13
Solution:
The solution to the system of linear equations is x = 7/13 and y = -5/13.
Therefore, we have successfully solved the system of linear equations 3x - 5y - 4 = 0 and 9x = 2y + 7 using the substitution method.
