Solving Linear Equations: 11v - 7u = 1 and 9v - 4u = 6
In this article, we will explore how to solve a system of linear equations, specifically the equations 11v - 7u = 1 and 9v - 4u = 6.
Understanding the Equations
Before we dive into solving the equations, let's take a moment to understand what each equation represents.
- The first equation, 11v - 7u = 1, indicates that 11 times the value of v minus 7 times the value of u is equal to 1.
- The second equation, 9v - 4u = 6, indicates that 9 times the value of v minus 4 times the value of u is equal to 6.
Our goal is to find the values of v and u that satisfy both equations.
Method of Substitution
One way to solve this system of equations is by using the method of substitution. Here's how:
- Solve one equation for one variable: Let's solve the first equation for v:
11v - 7u = 1 11v = 1 + 7u v = (1 + 7u) / 11
- Substitute the expression into the other equation: Now, substitute the expression for v into the second equation:
9v - 4u = 6 9((1 + 7u) / 11) - 4u = 6
- Simplify and solve for u: Simplify the equation and solve for u:
(9 + 63u) / 11 - 4u = 6 9 + 63u - 44u = 66 19u = 57 u = 57 / 19 u = 3
- Find the value of v: Now that we have the value of u, substitute it into one of the original equations to find the value of v:
v = (1 + 7u) / 11 v = (1 + 7(3)) / 11 v = (1 + 21) / 11 v = 22 / 11 v = 2
Solution
Therefore, the values of v and u that satisfy both equations are:
v = 2 u = 3
By plugging these values back into the original equations, we can verify that they satisfy both equations.
Conclusion
In this article, we used the method of substitution to solve the system of linear equations 11v - 7u = 1 and 9v - 4u = 6. We found that the values of v and u that satisfy both equations are v = 2 and u = 3.
