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Math  /  Algebra

Question5. Analyze each system. How would you rewrite the syste a. {12x−5y=−45−12x+10y=−20\left\{\begin{array}{c}\frac{1}{2} x-5 y=-45 \\ -\frac{1}{2} x+10 y=-20\end{array}\right. b. {4x+3y=243x+y=−2\left\{\begin{array}{l}4 x+3 y=24 \\ 3 x+y=-2\end{array}\right. c. {3x+5y=172x+3y=11\left\{\begin{array}{l}3 x+5 y=17 \\ 2 x+3 y=11\end{array}\right. d. 6x+3y=56 x+3 y=5 2x+y=12 x+y=1 {(3x+5y)=17)3(2x+3y=11)−5\left\{\begin{array}{l} (3 x+5 y)=17) 3 \\ (2 x+3 y=11)-5 \end{array}\right. e. {x+2y=−62x+4y=−12\left\{\begin{array}{c} x+2 y=-6 \\ 2 x+4 y=-12 \end{array}\right.

Studdy Solution

STEP 1

1. We are given systems of linear equations to analyze.
2. The goal is to rewrite each system in a form that might make solving easier.
3. We will use techniques such as simplifying equations, eliminating fractions, and identifying dependent or inconsistent systems.

STEP 2

1. Analyze and simplify system (a).
2. Analyze and simplify system (b).
3. Analyze and simplify system (c).
4. Analyze and simplify system (d).
5. Analyze and simplify system (e).

STEP 3

For system (a): {12x−5y=−45−12x+10y=−20\left\{ \begin{array}{c} \frac{1}{2} x - 5y = -45 \\ -\frac{1}{2} x + 10y = -20 \end{array} \right.
Multiply each equation by 2 to eliminate fractions:
{x−10y=−90−x+20y=−40\left\{ \begin{array}{c} x - 10y = -90 \\ -x + 20y = -40 \end{array} \right.

STEP 4

For system (b): {4x+3y=243x+y=−2\left\{ \begin{array}{l} 4x + 3y = 24 \\ 3x + y = -2 \end{array} \right.
This system is already in a simple form. Consider using substitution or elimination to solve.

STEP 5

For system (c): {3x+5y=172x+3y=11\left\{ \begin{array}{l} 3x + 5y = 17 \\ 2x + 3y = 11 \end{array} \right.
This system is also in a simple form. Consider using substitution or elimination to solve.

STEP 6

For system (d): {6x+3y=52x+y=1\left\{ \begin{array}{l} 6x + 3y = 5 \\ 2x + y = 1 \end{array} \right.
This system is already in a simple form. Consider using substitution or elimination to solve.

STEP 7

For system (e): {x+2y=−62x+4y=−12\left\{ \begin{array}{c} x + 2y = -6 \\ 2x + 4y = -12 \end{array} \right.
Notice that the second equation is a multiple of the first. This indicates that the system is dependent, meaning it has infinitely many solutions.
The systems have been analyzed and rewritten where necessary.

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