Solve each system of equations. If the system has no...

Solve each system of equations. If the system has no solution, say that it is inconsistent. For Problems 19-30, graph the lines of the system.
$$
\left\{\begin{array}{l}
\frac{1}{3} x-\frac{3}{2} y=-5 \\
\frac{3}{4} x+\frac{1}{3} y=11
\end{array}\right.
$$

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Systems of Linear Equations

A system of linear equations consists of two or more linear equations with the same set of variables. The solution to the system is the set of variable values that satisfy all equations simultaneously. In two-variable systems, the solution often represents the point where two lines intersect, if at all.

Graphical Representation of Linear Equations

Graphing involves plotting each linear equation on the coordinate plane as a line. By examining where the lines intersect, one can determine the nature of the system: a single intersection point indicates a unique solution, parallel lines show that there is no solution, and coinciding lines reveal infinitely many solutions.

Consistent and Inconsistent Systems

A consistent system has at least one solution, meaning the lines intersect at one point (or are the same line in the case of infinitely many solutions). An inconsistent system, on the other hand, has no solution because the lines are parallel and never intersect.

Algebraic Methods for Solving Systems

To solve systems of linear equations algebraically, methods such as substitution and elimination are commonly used. These methods involve manipulating the equations to isolate one variable or combine equations in a way that reduces the number of variables, allowing for the determination of the solution set.

Transcript

Please give Ace some feedback