## Quick Math System of Equations

##### Enter expression in this form

Example (two variables): 2x+5y =13, 8x-y=4

Example (three variables): 2x+9y+2z=3, 8x-9y+9z=4, 3x+7y-6z=9

Use this calculator to get quick answer on system of equations for two and three variables.

### Systems of Equations in Algebra

A system of equations is a set of two or more equations with the same variables. The solution to the system is the set of values that satisfy all equations simultaneously. There are several methods to solve systems of equations, including graphing, substitution, elimination, and matrix methods.

#### Methods to Solve Systems of Equations:

1. Graphing: Plotting each equation on a graph and finding the point(s) where they intersect.

2. Substitution: Solving one equation for one variable and substituting this expression into the other equation.

3. Elimination (Addition/Subtraction): Adding or subtracting equations to eliminate one of the variables.

4. Matrix Methods (Gauss-Jordan Elimination): Using matrices and row operations to solve the system.

#### Detailed Example:

Solve the following system of equations using the elimination method:

\[

\begin{cases}

2x + 3y = 13 \\

4x – y = 5

\end{cases}

\]

**1. Eliminate one variable:**

– Multiply the second equation by 3 to make the coefficients of \( y \) opposites:

\[

3(4x – y) = 3 \cdot 5 \implies 12x – 3y = 15

\]

**2. Add the equations:**

\[

\begin{array}{c}

2x + 3y = 13 \\

+ \, 12x – 3y = 15 \\

\hline

14x = 28

\end{array}

\]

**3. Solve for \( x \):**

\[

14x = 28 \implies x = 2

\]

**4. Substitute \( x = 2 \) back into one of the original equations to solve for \( y \):**

– Using the first equation:

\[

2(2) + 3y = 13 \implies 4 + 3y = 13 \implies 3y = 9 \implies y = 3

\]

So, the solution to the system is \( x = 2 \) and \( y = 3 \).

#### Verification:

Substitute \( x = 2 \) and \( y = 3 \) into both original equations to verify:

1. \( 2(2) + 3(3) = 4 + 9 = 13 \) (True)

2. \( 4(2) – 3 = 8 – 3 = 5 \) (True)

Both equations are satisfied, confirming that the solution is correct: \( x = 2 \) and \( y = 3 \).