Solved on Jan 12, 2024

Find the values of $x$ and $y$ that satisfy the system of linear equations $2x + 3y = 8$ and $x = \frac{1}{2}y$ .

STEP 1

Assumptions
1. We have a system of two linear equations: $2x + 3y = 8$ $x = \frac{1}{2}y$
2. We need to find the values of $x$ and $y$ that satisfy both equations simultaneously.

STEP 2

First, we express $x$ in terms of $y$ using the second equation.
$x = \frac{1}{2}y$

STEP 3

Now, we substitute the expression for $x$ from the second equation into the first equation.
$2\left(\frac{1}{2}y\right) + 3y = 8$

STEP 4

Simplify the equation by multiplying through.
$y + 3y = 8$

STEP 5

Combine like terms to get a single equation in $y$ .
$4y = 8$

STEP 6

Divide both sides of the equation by 4 to solve for $y$ .
$y = \frac{8}{4}$

STEP 7

Calculate the value of $y$ .
$y = 2$

STEP 8

Now that we have the value of $y$ , we can substitute it back into the second equation to find $x$ .
$x = \frac{1}{2}y$

STEP 9

Substitute the value of $y$ into the equation.
$x = \frac{1}{2}(2)$

STEP 10

Calculate the value of $x$ .
$x = 1$
The solution to the system of equations is $x = 1$ and $y = 2$ .

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Find the values of $x$ and $y$ that satisfy the system of linear equations $2x + 3y = 8$ and $x = \frac{1}{2}y$ .

STEP 1

Assumptions
1. We have a system of two linear equations: $2x + 3y = 8$ $x = \frac{1}{2}y$
2. We need to find the values of $x$ and $y$ that satisfy both equations simultaneously.

STEP 2

First, we express $x$ in terms of $y$ using the second equation.
$x = \frac{1}{2}y$

STEP 3

Now, we substitute the expression for $x$ from the second equation into the first equation.
$2\left(\frac{1}{2}y\right) + 3y = 8$

STEP 4

Simplify the equation by multiplying through.
$y + 3y = 8$

STEP 5

Combine like terms to get a single equation in $y$ .
$4y = 8$

STEP 6

Divide both sides of the equation by 4 to solve for $y$ .
$y = \frac{8}{4}$

STEP 7

Calculate the value of $y$ .
$y = 2$

STEP 8

Now that we have the value of $y$ , we can substitute it back into the second equation to find $x$ .
$x = \frac{1}{2}y$

STEP 9

Substitute the value of $y$ into the equation.
$x = \frac{1}{2}(2)$

STEP 10

Calculate the value of $x$ .
$x = 1$
The solution to the system of equations is $x = 1$ and $y = 2$ .

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Find the values of xxx and yyy that satisfy the system of linear equations 2x+3y=82x + 3y = 82x+3y=8 and x=12yx = \frac{1}{2}yx=21​y.

STEP 1

Assumptions1. We have a system of two linear equations: 2x+3y=82x + 3y = 82x+3y=8 x=12yx = \frac{1}{2}yx=21​y2. We need to find the values of xxx and yyy that satisfy both equations simultaneously.

STEP 2

First, we express xxx in terms of yyy using the second equation.x=12yx = \frac{1}{2}yx=21​y

STEP 3

Now, we substitute the expression for xxx from the second equation into the first equation.2(12y)+3y=82\left(\frac{1}{2}y\right) + 3y = 82(21​y)+3y=8

STEP 4

Simplify the equation by multiplying through.y+3y=8y + 3y = 8y+3y=8

STEP 5

Combine like terms to get a single equation in yyy.4y=84y = 84y=8

STEP 6

Divide both sides of the equation by 4 to solve for yyy.y=84y = \frac{8}{4}y=48​

STEP 7

Calculate the value of yyy.y=2y = 2y=2

STEP 8

Now that we have the value of yyy, we can substitute it back into the second equation to find xxx.x=12yx = \frac{1}{2}yx=21​y

STEP 9

Substitute the value of yyy into the equation.x=12(2)x = \frac{1}{2}(2)x=21​(2)

STEP 10

Calculate the value of xxx.x=1x = 1x=1The solution to the system of equations is x=1x = 1x=1 and y=2y = 2y=2.

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Find the values of $x$ and $y$ that satisfy the system of linear equations $2x + 3y = 8$ and $x = \frac{1}{2}y$ .

Assumptions
1. We have a system of two linear equations: $2x + 3y = 8$ $x = \frac{1}{2}y$
2. We need to find the values of $x$ and $y$ that satisfy both equations simultaneously.

First, we express $x$ in terms of $y$ using the second equation.
$x = \frac{1}{2}y$

Now, we substitute the expression for $x$ from the second equation into the first equation.
$2\left(\frac{1}{2}y\right) + 3y = 8$

Simplify the equation by multiplying through.
$y + 3y = 8$

Combine like terms to get a single equation in $y$ .
$4y = 8$

Divide both sides of the equation by 4 to solve for $y$ .
$y = \frac{8}{4}$

Calculate the value of $y$ .
$y = 2$

Now that we have the value of $y$ , we can substitute it back into the second equation to find $x$ .
$x = \frac{1}{2}y$

Substitute the value of $y$ into the equation.
$x = \frac{1}{2}(2)$

Calculate the value of $x$ .
$x = 1$
The solution to the system of equations is $x = 1$ and $y = 2$ .