Question

solve the following system of linear equation algebraically
ax=by
3ax-2by=ab

Answer 1

Have you considered attempting the solution?

Answer 2

I have tried to but it ended up confusing me more

Answer 3

by substitution of the first equation into the second one, we get:
\[\begin{gathered}
3by - 2by = ab \hfill \\
\hfill \\
by = ab \hfill \\
\end{gathered} \]

Answer 4

now, we have to distinguish two cases, namely:
1) \(b \neq 0\)
so we can divide both sides by \(b\), and we get:
\[\frac{{by}}{b} = \frac{{ab}}{b}\]
please continue

Answer 5

what is \(y\)?

Answer 6

y=a?

Answer 7

perfect!

Answer 8

now I substitute \(y=a\) into the first equation, and I get:
\[ax = ab\]

Answer 9

again I have to consider two cases, namely:
1) \(a \neq 0\)
2) \(a=0\)
case 1)
being \(a \neq 0\) I can divide by \(a\), so I get:
\[\frac{{ax}}{a} = \frac{{ab}}{a}\]
what is \(x\) ?

Answer 10

x=b

Answer 11

correct!
so we have the first solution of the system, like below:
\[\left\{ \begin{gathered}
y = a \hfill \\
x = b \hfill \\
\end{gathered} \right.,\quad a \ne 0,b \ne 0\]

Answer 12

next we have to consider the case 2)
being \(a=0\), how many solution has this equations:
\[ax = ab\]

Answer 13

solutions*

Answer 14

1?

Answer 15

if \(a=0\), we can write:
\(0\;x=0\)
so any number satisfies such equation, right?

Answer 16

yes

Answer 17

so such equation has infinite solutions, and the second solution of the system is:
\[\left\{ \begin{gathered}
y = a \hfill \\
x = k \hfill \\
\end{gathered} \right.,\quad a = 0,b \ne 0,k \in \mathbb{R}\]

Answer 18

now we have to consider this case:
2) \(b=0\)

Answer 19

in that case I can rewrite the above equation:
\[by = ab\]
in this way:
\(0\;y=0\)
so how many solutions \(y\) has such equation?

Answer 20

infinite amount of solutions

Answer 21

correct!

Answer 22

we can write this:
\[y = h,\quad h \in \mathbb{R}\]

Answer 23

now, I substitute \(y=h\) into the original first equation:
\[ax = bk = 0\]
since \(b=0\)
then we have:
\(ax=0\)
Now, as befor, I have to distinguish two cases:
1) \(a \neq 0\)
2) \(a=0\)
now, I consider the case 1):
\(a \neq 0\)
so I can divide by \(a\), and I get:
\[\frac{{ax}}{a} = \frac{0}{a}\]
what is \(x\) ?

Answer 24

before*

Answer 25

\[x=\frac{ 0 }{ a }\]

Answer 26

which is equal to \(0\), so the third solution of the system is:
\[\left\{ \begin{gathered}
y = h \hfill \\
x = 0 \hfill \\
\end{gathered} \right.,\quad a \ne 0,b = 0,h \in \mathbb{R}\]

Answer 27

next I have to consider the case \(a=0\)
now I can rewrite the equation:
\[ax = 0\]
like below:
\(0\;x=0\)
so, how many solutions \(x\) has such equation

Answer 28

infinite amount of solutions

Answer 29

that's rright!
so we can write the fourth solution of the system as below:
\[\left\{ \begin{gathered}
y = h \hfill \\
x = m \hfill \\
\end{gathered} \right.,\quad a = 0,b = 0,\quad h,m \in \mathbb{R}\]

Answer 30

oops.. that's right*!

Answer 31

and we have completed your exercise

Answer 32

Create a system of equations that includes one linear equation and one quadratic equation.

Answer 33

Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.

Answer 34

HELP >> MEDAL GIVEN OUT
Create a single variable linear equation that has no solution.

Answer 35

Please help? I'm lost with these two questions. Fan + Medal for a reasonable answer.

Answer 36

*MEDAL*FAN*
Create a system of equations that includes one linear equation and one quadratic equation.

Answer 37

Solve a Linear System Graphically- can someone please explain how to solve these? I missed the lesson so I don't really understand how to do it.

Answer 38

Please solve the linear quadratic system algebraically
y=x^2-8x-12
y=4x+8

Answer 39

Solve the following system of linear equations algebraically
-2x+y=2
x+y=-1

Answer 40

College Algebra: I'm doing 2X2 Non-Linear Systems and I'm stuck on this problem. I need to solve this system algebraically and then graph both of the equati

Answer 41

NAFTA has been hugely debated by many people. Identify 3 pros, and 3 cons to this trilateral trade agreement.

Answer 42

Quick question =)

Answer 43

The use of phrases, fragments, and punctuation can affect the pace and mood of a text.

Answer 44

Nadia swims at a rate of 50 meters per minute. Create a function f, where f(n) gives the number of meters Nadia swims given the number of minutes she swims,

Answer 45

Nadia swims at a rate of 50 meters per minute. Create a function f, where f(n) gives the number of meters Nadia swims given the number of minutes she swims,

Answer 46

identify the most powerful aspects of gates mister jefferson and the trials of ph

Answer 47

Nadia swims at a rate of 50 meters per minute. Create a function f, where f(n) gives the number of meters Nadia swims given the number of minutes she swims,

Answer 48

Which Inequalities represent Susan's goal for earning money and the number of hours she can work in one week?\nSusan is a college student with two part-time

Answer 49

Testing

Answer 50

help please!

Answer 51

which of these lines does Sir Gawain display honesty and modesty by publicly ackn

Answer 52

Raul and Josephine buy a \nplasma TV from Sparky\u2019s Electronics on an installment plan \n(simple interest add on loan).