Finite Mathematik Beispiele

$\frac{2}{3} x + y = 7$ , $\frac{1}{2} x + \frac{3}{4} y = \frac{21}{4}$

Schritt 1

Move variables to the left and constant terms to the right.

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Schritt 1.1

Kombiniere $\frac{2}{3}$ und $x$ .

$\frac{2 x}{3} + y = 7$

$\frac{1}{2} x + \frac{3}{4} y = \frac{21}{4}$

Schritt 1.2

Stelle die Terme um.

$\frac{2}{3} x + y = 7$

$\frac{1}{2} x + \frac{3}{4} y = \frac{21}{4}$

Schritt 1.3

Vereinfache jeden Term.

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Schritt 1.3.1

Kombiniere $\frac{1}{2}$ und $x$ .

$\frac{2}{3} x + y = 7$

$\frac{x}{2} + \frac{3}{4} y = \frac{21}{4}$

Schritt 1.3.2

Kombiniere $\frac{3}{4}$ und $y$ .

$\frac{2}{3} x + y = 7$

$\frac{x}{2} + \frac{3 y}{4} = \frac{21}{4}$

$\frac{2}{3} x + y = 7$

$\frac{x}{2} + \frac{3 y}{4} = \frac{21}{4}$

Schritt 1.4

Stelle die Terme um.

$\frac{2}{3} x + y = 7$

$\frac{1}{2} x + \frac{3}{4} y = \frac{21}{4}$

$\frac{2}{3} x + y = 7$

$\frac{1}{2} x + \frac{3}{4} y = \frac{21}{4}$

Schritt 2

Write the system as a matrix.

$[\begin{array}{c} \frac{2}{3} & 1 & 7 \\ \frac{1}{2} & \frac{3}{4} & \frac{21}{4} \end{array}]$

Schritt 3

Ermittele die normierte Zeilenstufenform.

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Schritt 3.1

Multiply each element of $R_{1}$ by $\frac{3}{2}$ to make the entry at $1, 1$ a $1$ .

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Schritt 3.1.1

Multiply each element of $R_{1}$ by $\frac{3}{2}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{3}{2} \cdot \frac{2}{3} & \frac{3}{2} \cdot 1 & \frac{3}{2} \cdot 7 \\ \frac{1}{2} & \frac{3}{4} & \frac{21}{4} \end{array}]$

Schritt 3.1.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & \frac{3}{2} & \frac{21}{2} \\ \frac{1}{2} & \frac{3}{4} & \frac{21}{4} \end{array}]$

Schritt 3.2

Perform the row operation $R_{2} = R_{2} - \frac{1}{2} R_{1}$ to make the entry at $2, 1$ a $0$ .

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Schritt 3.2.1

Perform the row operation $R_{2} = R_{2} - \frac{1}{2} R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{3}{2} & \frac{21}{2} \\ \frac{1}{2} - \frac{1}{2} \cdot 1 & \frac{3}{4} - \frac{1}{2} \cdot \frac{3}{2} & \frac{21}{4} - \frac{1}{2} \cdot \frac{21}{2} \end{array}]$

Schritt 3.2.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & \frac{3}{2} & \frac{21}{2} \\ 0 & 0 & 0 \end{array}]$

Schritt 4

Use the result matrix to declare the final solution to the system of equations.

$x + \frac{3}{2} y = \frac{21}{2}$

$0 = 0$

Schritt 5

The solution is the set of ordered pairs that make the system true.

$(\frac{21}{2} - \frac{3 y}{2}, y)$