Solve each system. If a system is inconsistent or if the equations are dependent, state this. {

Solve each system. If a system is inconsistent or if the...

Key Concepts

System of Linear Equations

This concept involves a set of equations where each equation is linear in the unknowns. The goal is to find values for the variables that satisfy all equations simultaneously. It forms the foundation for many problems in algebra, with real-world applications in areas such as engineering and economics.

Consistency and Inconsistency

A system is consistent if there exists at least one solution that satisfies all equations; it is inconsistent if no solution exists. Recognizing consistency helps determine whether to proceed with finding a unique solution, multiple solutions, or concluding that no solution exists.

Dependent and Independent Equations

Dependent equations are multiple representations of the same relationship, leading to an infinite number of solutions if the system is consistent. Independent equations, on the other hand, provide unique information resulting in a single solution if the system is consistent.

Elimination Method

This method involves combining equations to cancel out one or more of the variables, simplifying the system into fewer equations. It is a powerful algebraic technique used to systematically reduce a system, making it easier to solve for the remaining variables.

Substitution Method

This method entails solving one of the equations for one variable and then substituting that expression into the other equations. It reduces the number of variables in the equations, often making the system simpler to solve step-by-step.

Transcript

00:01 Okay, so to begin this, i'm going to use elimination and try to get rid of c using the first equation and the second equation in the problem.

00:12 So with that, i have 7a plus 9b minus 2c is equal to negative 5.

00:22 So that's my first equation.

00:23 My second equation is 5a plus 14b minus c is equal to negative 11.

00:32 So i'm going to now multiply the second equation here by negative 2 so i can eliminate c.

00:39 So with that, my two equations now line up to be 7a plus 9b minus 2c is equal to negative 5.

00:48 And my second line, so when i distribute the negative 2 to this whole equation, i'll have negative 10a minus 28b plus 2c is equal to negative 22.

01:04 Or positive 22.

01:07 Okay, so here, if i now add these going down, these two terms drop out.

01:13 So with that, i'll have negative 3a minus 19b is equal to negative 17.

01:21 So i'm going to hold on to this equation that i have here.

01:25 So now i'm going to try and eliminate.

01:29 I'm going to eliminate c using the second and third equation, just like we did in the first part.

01:35 So with that, i have 5a plus 14b minus c is equal to negative 11.

01:43 So that's my second equation.

01:45 And then 2a minus 5b minus c is equal to positive 3...