Math / Algebra

Question $\begin{array}{l}20 x-5 y=5 \\ 4 x-y=1\end{array}$

Studdy Solution

STEP 1

What is this asking? We're trying to find the mystery values of $x$ and $y$ that make both of these equations true at the same time! Watch out! Don't forget, there are two equations here, and our $x$ and $y$ values need to satisfy both of them.
It's a team effort!

STEP 2

1. Simplify the first equation
2. Express $y$ in terms of $x$
3. Substitute and solve for $x$
4. Substitute and solve for $y$

STEP 3

Look at that first equation, $20x - 5y = 5$ .
Notice how all those numbers are divisible by 5?
Let's make things easier by dividing both sides by 5!
Remember, what we do to one side, we must do to the other to keep the equation balanced.
This gives us $4x - y = 1$ .
Much cleaner!

STEP 4

Now, let's get that $y$ by itself in our simplified equation.
We can do this by adding $y$ to both sides and subtracting $1$ from both sides.
This gives us $y = 4x - 1$ .
Now we know what $y$ is in terms of $x$ !

STEP 5

Remember our second equation, $4x - y = 1$ ?
Let's swap out the $y$ with what we just found: $4x - 1$ .
This gives us $4x - (4x - 1) = 1$ .

STEP 6

Now, distribute that negative sign: $4x - 4x + 1 = 1$ .
Whoa, the $4x$ terms add to zero!
We're left with $1 = 1$ .
This means that any value of $x$ will work!

STEP 7

Since any $x$ works, let's pick a simple one: $x = 0$ .

STEP 8

Plug $x = 0$ into our equation for $y$ : $y = 4 \cdot 0 - 1$ , which simplifies to $y = -1$ .

STEP 9

Just to be sure, let's try $x = 1$ .
Then $y = 4 \cdot 1 - 1 = 3$ .

STEP 10

There are infinitely many solutions!
Since any $x$ value works, we can express the solution as a set of ordered pairs: $(x, 4x - 1)$ .
For example, when $x = 0$ , $y = -1$ , and when $x = 1$ , $y = 3$ .

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Question $\begin{array}{l}20 x-5 y=5 \\ 4 x-y=1\end{array}$

Studdy Solution

STEP 1

What is this asking? We're trying to find the mystery values of $x$ and $y$ that make both of these equations true at the same time! Watch out! Don't forget, there are two equations here, and our $x$ and $y$ values need to satisfy both of them.
It's a team effort!

STEP 2

1. Simplify the first equation
2. Express $y$ in terms of $x$
3. Substitute and solve for $x$
4. Substitute and solve for $y$

STEP 3

Look at that first equation, $20x - 5y = 5$ .
Notice how all those numbers are divisible by 5?
Let's make things easier by dividing both sides by 5!
Remember, what we do to one side, we must do to the other to keep the equation balanced.
This gives us $4x - y = 1$ .
Much cleaner!

STEP 4

Now, let's get that $y$ by itself in our simplified equation.
We can do this by adding $y$ to both sides and subtracting $1$ from both sides.
This gives us $y = 4x - 1$ .
Now we know what $y$ is in terms of $x$ !

STEP 5

Remember our second equation, $4x - y = 1$ ?
Let's swap out the $y$ with what we just found: $4x - 1$ .
This gives us $4x - (4x - 1) = 1$ .

STEP 6

Now, distribute that negative sign: $4x - 4x + 1 = 1$ .
Whoa, the $4x$ terms add to zero!
We're left with $1 = 1$ .
This means that any value of $x$ will work!

STEP 7

Since any $x$ works, let's pick a simple one: $x = 0$ .

STEP 8

Plug $x = 0$ into our equation for $y$ : $y = 4 \cdot 0 - 1$ , which simplifies to $y = -1$ .

STEP 9

Just to be sure, let's try $x = 1$ .
Then $y = 4 \cdot 1 - 1 = 3$ .

STEP 10

There are infinitely many solutions!
Since any $x$ value works, we can express the solution as a set of ordered pairs: $(x, 4x - 1)$ .
For example, when $x = 0$ , $y = -1$ , and when $x = 1$ , $y = 3$ .

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Question20x−5y=54x−y=1\begin{array}{l}20 x-5 y=5 \\ 4 x-y=1\end{array}20x−5y=54x−y=1​

Studdy Solution

STEP 1

What is this asking? We're trying to find the mystery values of xxx and yyy that make *both* of these equations true at the same time! Watch out! Don't forget, there are *two* equations here, and our xxx and yyy values need to satisfy both of them.It's a team effort!

STEP 2

1. Simplify the first equation2. Express yyy in terms of xxx3. Substitute and solve for xxx4. Substitute and solve for yyy

STEP 3

STEP 4

Now, let's get that yyy by itself in our simplified equation.We can do this by adding yyy to both sides and subtracting 111 from both sides.This gives us y=4x−1y = 4x - 1y=4x−1.Now we know what yyy is in terms of xxx!

STEP 5

Remember our second equation, 4x−y=14x - y = 14x−y=1?Let's swap out the yyy with what we just found: 4x−14x - 14x−1.This gives us 4x−(4x−1)=14x - (4x - 1) = 14x−(4x−1)=1.

STEP 6

Now, distribute that negative sign: 4x−4x+1=14x - 4x + 1 = 14x−4x+1=1.Whoa, the 4x4x4x terms add to zero!We're left with 1=11 = 11=1.This means that *any* value of xxx will work!

STEP 7

Since *any* xxx works, let's pick a simple one: x=0x = 0x=0.

STEP 8

Plug x=0x = 0x=0 into our equation for yyy: y=4⋅0−1y = 4 \cdot 0 - 1y=4⋅0−1, which simplifies to y=−1y = -1y=−1.

STEP 9

Just to be sure, let's try x=1x = 1x=1.Then y=4⋅1−1=3y = 4 \cdot 1 - 1 = 3y=4⋅1−1=3.

STEP 10

There are infinitely many solutions!Since any xxx value works, we can express the solution as a set of ordered pairs: (x,4x−1)(x, 4x - 1)(x,4x−1).For example, when x=0x = 0x=0, y=−1y = -1y=−1, and when x=1x = 1x=1, y=3y = 3y=3.

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Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Question $\begin{array}{l}20 x-5 y=5 \\ 4 x-y=1\end{array}$

What is this asking? We're trying to find the mystery values of $x$ and $y$ that make both of these equations true at the same time! Watch out! Don't forget, there are two equations here, and our $x$ and $y$ values need to satisfy both of them.
It's a team effort!

1. Simplify the first equation
2. Express $y$ in terms of $x$
3. Substitute and solve for $x$
4. Substitute and solve for $y$

Now, let's get that $y$ by itself in our simplified equation.
We can do this by adding $y$ to both sides and subtracting $1$ from both sides.
This gives us $y = 4x - 1$ .
Now we know what $y$ is in terms of $x$ !

Remember our second equation, $4x - y = 1$ ?
Let's swap out the $y$ with what we just found: $4x - 1$ .
This gives us $4x - (4x - 1) = 1$ .

Now, distribute that negative sign: $4x - 4x + 1 = 1$ .
Whoa, the $4x$ terms add to zero!
We're left with $1 = 1$ .
This means that any value of $x$ will work!

Since any $x$ works, let's pick a simple one: $x = 0$ .

Plug $x = 0$ into our equation for $y$ : $y = 4 \cdot 0 - 1$ , which simplifies to $y = -1$ .

Just to be sure, let's try $x = 1$ .
Then $y = 4 \cdot 1 - 1 = 3$ .

There are infinitely many solutions!
Since any $x$ value works, we can express the solution as a set of ordered pairs: $(x, 4x - 1)$ .
For example, when $x = 0$ , $y = -1$ , and when $x = 1$ , $y = 3$ .