Solved on Dec 20, 2023

Solve the system of linear equations: $2w - x = 5$ and $4w = 2x + 10$ .

STEP 1

Assumptions
1. We have two equations: $2w - x = 5$ and $4w = 2x + 10$ .
2. We need to solve for the variables $w$ and $x$ .

STEP 2

First, we will solve one of the equations for one of the variables. Let's solve the first equation for $x$ .
$x = 2w - 5$

STEP 3

Now we will substitute the expression for $x$ from the first equation into the second equation to solve for $w$ .
$4w = 2(2w - 5) + 10$

STEP 4

Distribute the $2$ on the right side of the equation.
$4w = 4w - 10 + 10$

STEP 5

Notice that the terms $4w$ on both sides of the equation cancel each other out. We are left with:
$0 = 0$
This indicates that the two equations are dependent and represent the same line. There are infinitely many solutions for $w$ and $x$ that satisfy both equations.

STEP 6

To express the solution, we can use the expression for $x$ we found in STEP_2 and state that $w$ can be any real number.
The solution is:
$x = 2w - 5$ $w \in \mathbb{R}$

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Solve the system of linear equations: $2w - x = 5$ and $4w = 2x + 10$ .

STEP 1

Assumptions
1. We have two equations: $2w - x = 5$ and $4w = 2x + 10$ .
2. We need to solve for the variables $w$ and $x$ .

STEP 2

First, we will solve one of the equations for one of the variables. Let's solve the first equation for $x$ .
$x = 2w - 5$

STEP 3

Now we will substitute the expression for $x$ from the first equation into the second equation to solve for $w$ .
$4w = 2(2w - 5) + 10$

STEP 4

Distribute the $2$ on the right side of the equation.
$4w = 4w - 10 + 10$

STEP 5

Notice that the terms $4w$ on both sides of the equation cancel each other out. We are left with:
$0 = 0$
This indicates that the two equations are dependent and represent the same line. There are infinitely many solutions for $w$ and $x$ that satisfy both equations.

STEP 6

To express the solution, we can use the expression for $x$ we found in STEP_2 and state that $w$ can be any real number.
The solution is:
$x = 2w - 5$ $w \in \mathbb{R}$

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Solve the system of linear equations: 2w−x=52w - x = 52w−x=5 and 4w=2x+104w = 2x + 104w=2x+10.

STEP 1

Assumptions1. We have two equations: 2w−x=52w - x = 52w−x=5 and 4w=2x+104w = 2x + 104w=2x+10.2. We need to solve for the variables www and xxx.

STEP 2

First, we will solve one of the equations for one of the variables. Let's solve the first equation for xxx.x=2w−5x = 2w - 5x=2w−5

STEP 3

Now we will substitute the expression for xxx from the first equation into the second equation to solve for www.4w=2(2w−5)+104w = 2(2w - 5) + 104w=2(2w−5)+10

STEP 4

Distribute the 222 on the right side of the equation.4w=4w−10+104w = 4w - 10 + 104w=4w−10+10

STEP 5

Notice that the terms 4w4w4w on both sides of the equation cancel each other out. We are left with:0=00 = 00=0This indicates that the two equations are dependent and represent the same line. There are infinitely many solutions for www and xxx that satisfy both equations.

STEP 6

To express the solution, we can use the expression for xxx we found in STEP_2 and state that www can be any real number.The solution is:x=2w−5x = 2w - 5x=2w−5 w∈Rw \in \mathbb{R}w∈R

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

Solve the system of linear equations: $2w - x = 5$ and $4w = 2x + 10$ .

Assumptions
1. We have two equations: $2w - x = 5$ and $4w = 2x + 10$ .
2. We need to solve for the variables $w$ and $x$ .

First, we will solve one of the equations for one of the variables. Let's solve the first equation for $x$ .
$x = 2w - 5$

Now we will substitute the expression for $x$ from the first equation into the second equation to solve for $w$ .
$4w = 2(2w - 5) + 10$

Distribute the $2$ on the right side of the equation.
$4w = 4w - 10 + 10$

Notice that the terms $4w$ on both sides of the equation cancel each other out. We are left with:
$0 = 0$
This indicates that the two equations are dependent and represent the same line. There are infinitely many solutions for $w$ and $x$ that satisfy both equations.

To express the solution, we can use the expression for $x$ we found in STEP_2 and state that $w$ can be any real number.
The solution is:
$x = 2w - 5$ $w \in \mathbb{R}$