91Ó°ÊÓ

A system of equations is a collection of two or more equations with the same set of variables. In our exercise, we have two equations: the first one is \(6x - y = 5 \) and the second one is \( y = 11x \). To solve a system of equations, we need to find the values of the variables that satisfy all equations in the system simultaneously. There are different methods to solve such systems, including graphing, substitution, and elimination. In this exercise, we use the substitution method. Always remember, solutions to a system of equations can either be a single point, infinitely many solutions, or no solutions at all.

Solving linear equations means finding the value(s) of the variable(s) that make the equation true. Linear equations are equations of the first degree, meaning they have variables raised only to the power of one. In the given exercise: the equations \(6x - y = 5 \) and \( y = 11x \) are both linear. The goal is to find the values of \( x \) and \( y \) which satisfy both equations. By substituting the expression from the second equation into the first, we reduce the system to a single linear equation in one variable. Once simplified, we solve this equation to find \( x \). After finding \( x \), we substitute it back into one of the original equations to find the corresponding value of \( y \).

Substitution is a method used to solve a system of equations by replacing one variable with an expression derived from another equation. This method simplifies the system step by step. Here's how we applied substitution to our specific problem:

• First, we identified the second equation, which is already solved for \( y \): \( y = 11x \).
• We then substituted this value of \( y \) into the first equation to get: \( 6x - 11x = 5 \).
• Next, we simplified the equation to solve for \( x \): \( - 5x = 5 \).
• Dividing both sides by -5, we found \( x = -1 \).

After finding \( x \), we substituted \( x \) back into the second equation to find \( y \):

• Substituting \( x = -1 \) into \( y = 11x \), we get \( y = 11(-1) = -11 \).

Thus, the solution to the system is \( x = -1 \) and \( y = -11 \). Substitution is particularly useful when one of the equations is already solved for one variable.