Solved on Jan 04, 2024

Solve the system of linear equations $5x - y = 5$ and $5x + y = 5$ .

STEP 1

Assumptions
1. We have a system of two linear equations with two variables, $x$ and $y$ .
2. The system is given by: \begin{align} 5x - y &= 5, \\ 5x + y &= 5. \end{align}
3. We aim to find the values of $x$ and $y$ that satisfy both equations simultaneously.

STEP 2

To solve the system, we can add the two equations together to eliminate $y$ .
$(5x - y) + (5x + y) = 5 + 5$

STEP 3

Perform the addition of the left-hand sides of the equations.
$5x - y + 5x + y = 10x$

STEP 4

Perform the addition of the right-hand sides of the equations.
$5 + 5 = 10$

STEP 5

Combine the results from STEP_3 and STEP_4 to form a new equation.
$10x = 10$

STEP 6

Solve for $x$ by dividing both sides of the equation by 10.
$x = \frac{10}{10}$

STEP 7

Calculate the value of $x$ .
$x = 1$

STEP 8

Now that we have the value of $x$ , we can substitute it back into either of the original equations to solve for $y$ . We will use the first equation.
$5x - y = 5$

STEP 9

Substitute $x = 1$ into the equation.
$5(1) - y = 5$

STEP 10

Perform the multiplication.
$5 - y = 5$

STEP 11

To solve for $y$ , we can subtract 5 from both sides of the equation.
$-y = 5 - 5$

STEP 12

Calculate the right-hand side of the equation.
$-y = 0$

STEP 13

Multiply both sides of the equation by $-1$ to solve for $y$ .
$y = 0$
The solution to the system of equations is $x = 1$ and $y = 0$ .

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Solve the system of linear equations $5x - y = 5$ and $5x + y = 5$ .

STEP 1

Assumptions
1. We have a system of two linear equations with two variables, $x$ and $y$ .
2. The system is given by: \begin{align} 5x - y &= 5, \\ 5x + y &= 5. \end{align}
3. We aim to find the values of $x$ and $y$ that satisfy both equations simultaneously.

STEP 2

To solve the system, we can add the two equations together to eliminate $y$ .
$(5x - y) + (5x + y) = 5 + 5$

STEP 3

Perform the addition of the left-hand sides of the equations.
$5x - y + 5x + y = 10x$

STEP 4

Perform the addition of the right-hand sides of the equations.
$5 + 5 = 10$

STEP 5

Combine the results from STEP_3 and STEP_4 to form a new equation.
$10x = 10$

STEP 6

Solve for $x$ by dividing both sides of the equation by 10.
$x = \frac{10}{10}$

STEP 7

Calculate the value of $x$ .
$x = 1$

STEP 8

Now that we have the value of $x$ , we can substitute it back into either of the original equations to solve for $y$ . We will use the first equation.
$5x - y = 5$

STEP 9

Substitute $x = 1$ into the equation.
$5(1) - y = 5$

STEP 10

Perform the multiplication.
$5 - y = 5$

STEP 11

To solve for $y$ , we can subtract 5 from both sides of the equation.
$-y = 5 - 5$

STEP 12

Calculate the right-hand side of the equation.
$-y = 0$

STEP 13

Multiply both sides of the equation by $-1$ to solve for $y$ .
$y = 0$
The solution to the system of equations is $x = 1$ and $y = 0$ .

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Solve the system of linear equations 5x−y=55x - y = 55x−y=5 and 5x+y=55x + y = 55x+y=5.

STEP 1

Assumptions1. We have a system of two linear equations with two variables, xxx and yyy.2. The system is given by: \begin{align*} 5x - y &= 5, \\ 5x + y &= 5. \end{align*}3. We aim to find the values of xxx and yyy that satisfy both equations simultaneously.

STEP 2

To solve the system, we can add the two equations together to eliminate yyy.(5x−y)+(5x+y)=5+5 (5x - y) + (5x + y) = 5 + 5 (5x−y)+(5x+y)=5+5

STEP 3

Perform the addition of the left-hand sides of the equations.5x−y+5x+y=10x 5x - y + 5x + y = 10x 5x−y+5x+y=10x

STEP 4

Perform the addition of the right-hand sides of the equations.5+5=10 5 + 5 = 10 5+5=10

STEP 5

Combine the results from STEP_3 and STEP_4 to form a new equation.10x=10 10x = 10 10x=10

STEP 6

Solve for xxx by dividing both sides of the equation by 10.x=1010 x = \frac{10}{10} x=1010​

STEP 7

Calculate the value of xxx.x=1 x = 1 x=1

STEP 8

Now that we have the value of xxx, we can substitute it back into either of the original equations to solve for yyy. We will use the first equation.5x−y=5 5x - y = 5 5x−y=5

STEP 9

Substitute x=1x = 1x=1 into the equation.5(1)−y=5 5(1) - y = 5 5(1)−y=5

STEP 10

Perform the multiplication.5−y=5 5 - y = 5 5−y=5

STEP 11

To solve for yyy, we can subtract 5 from both sides of the equation.−y=5−5 -y = 5 - 5 −y=5−5

STEP 12

Calculate the right-hand side of the equation.−y=0 -y = 0 −y=0

STEP 13

Multiply both sides of the equation by −1-1−1 to solve for yyy.y=0 y = 0 y=0The solution to the system of equations is x=1x = 1x=1 and y=0y = 0y=0.

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Solve the system of linear equations $5x - y = 5$ and $5x + y = 5$ .

Assumptions
1. We have a system of two linear equations with two variables, $x$ and $y$ .
2. The system is given by: \begin{align} 5x - y &= 5, \\ 5x + y &= 5. \end{align}
3. We aim to find the values of $x$ and $y$ that satisfy both equations simultaneously.

To solve the system, we can add the two equations together to eliminate $y$ .
$(5x - y) + (5x + y) = 5 + 5$

Perform the addition of the left-hand sides of the equations.
$5x - y + 5x + y = 10x$

Perform the addition of the right-hand sides of the equations.
$5 + 5 = 10$

Combine the results from STEP_3 and STEP_4 to form a new equation.
$10x = 10$

Solve for $x$ by dividing both sides of the equation by 10.
$x = \frac{10}{10}$

Calculate the value of $x$ .
$x = 1$

Now that we have the value of $x$ , we can substitute it back into either of the original equations to solve for $y$ . We will use the first equation.
$5x - y = 5$

Substitute $x = 1$ into the equation.
$5(1) - y = 5$

Perform the multiplication.
$5 - y = 5$

To solve for $y$ , we can subtract 5 from both sides of the equation.
$-y = 5 - 5$

Calculate the right-hand side of the equation.
$-y = 0$

Multiply both sides of the equation by $-1$ to solve for $y$ .
$y = 0$
The solution to the system of equations is $x = 1$ and $y = 0$ .