Question Solve the system of linear equations to find the common solution.
$x+y=20$ and $y=2x-1$ $y=2x$ and $y=-6x+4$

Studdy Solution

STEP 1

Assumptions
1. We are given two systems of linear equations.
2. Each system has two equations with two variables, $x$ and $y$ .
3. We need to find the common solution for each system, which means finding the values of $x$ and $y$ that satisfy both equations in the system.

STEP 2

Solve the first system of equations:
$\begin{array}{l} x + y = 20 \\ y = 2x - 1 \end{array}$

STEP 3

Substitute the expression for $y$ from the second equation into the first equation.
$x + (2x - 1) = 20$

STEP 4

Combine like terms in the equation.
$3x - 1 = 20$

STEP 5

Add 1 to both sides of the equation to isolate the term with $x$ .
$3x = 21$

STEP 6

Divide both sides by 3 to solve for $x$ .
$x = 7$

STEP 7

Now that we have the value of $x$ , we can substitute it back into the second equation to solve for $y$ .
$y = 2(7) - 1$

STEP 8

Calculate the value of $y$ .
$y = 14 - 1$
$y = 13$

STEP 9

The common solution for the first system of equations is $x = 7$ and $y = 13$ .

STEP 10

Now, solve the second system of equations:
$\begin{array}{l} y = 2x \\ y = -6x + 4 \end{array}$

STEP 11

Since both equations equal $y$ , we can set them equal to each other to find $x$ .
$2x = -6x + 4$

STEP 12

Add $6x$ to both sides of the equation to get all $x$ terms on one side.
$8x = 4$

STEP 13

Divide both sides by 8 to solve for $x$ .
$x = \frac{4}{8}$

STEP 14

Simplify the fraction to find the value of $x$ .
$x = \frac{1}{2}$

STEP 15

Now that we have the value of $x$ , we can substitute it back into either of the original equations to solve for $y$ . We'll use the first equation.
$y = 2 \left(\frac{1}{2}\right)$

STEP 16

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

STEP 17

The common solution for the second system of equations is $x = \frac{1}{2}$ and $y = 1$ .
The solutions to the systems of equations are:
For system 7: $x = 7, y = 13$
For system 8: $x = \frac{1}{2}, y = 1$

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Question Solve the system of linear equations to find the common solution.
$x+y=20$ and $y=2x-1$ $y=2x$ and $y=-6x+4$

Studdy Solution

STEP 1

Assumptions
1. We are given two systems of linear equations.
2. Each system has two equations with two variables, $x$ and $y$ .
3. We need to find the common solution for each system, which means finding the values of $x$ and $y$ that satisfy both equations in the system.

STEP 2

Solve the first system of equations:
$\begin{array}{l} x + y = 20 \\ y = 2x - 1 \end{array}$

STEP 3

Substitute the expression for $y$ from the second equation into the first equation.
$x + (2x - 1) = 20$

STEP 4

Combine like terms in the equation.
$3x - 1 = 20$

STEP 5

Add 1 to both sides of the equation to isolate the term with $x$ .
$3x = 21$

STEP 6

Divide both sides by 3 to solve for $x$ .
$x = 7$

STEP 7

Now that we have the value of $x$ , we can substitute it back into the second equation to solve for $y$ .
$y = 2(7) - 1$

STEP 8

Calculate the value of $y$ .
$y = 14 - 1$
$y = 13$

STEP 9

The common solution for the first system of equations is $x = 7$ and $y = 13$ .

STEP 10

Now, solve the second system of equations:
$\begin{array}{l} y = 2x \\ y = -6x + 4 \end{array}$

STEP 11

Since both equations equal $y$ , we can set them equal to each other to find $x$ .
$2x = -6x + 4$

STEP 12

Add $6x$ to both sides of the equation to get all $x$ terms on one side.
$8x = 4$

STEP 13

Divide both sides by 8 to solve for $x$ .
$x = \frac{4}{8}$

STEP 14

Simplify the fraction to find the value of $x$ .
$x = \frac{1}{2}$

STEP 15

Now that we have the value of $x$ , we can substitute it back into either of the original equations to solve for $y$ . We'll use the first equation.
$y = 2 \left(\frac{1}{2}\right)$

STEP 16

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

STEP 17

The common solution for the second system of equations is $x = \frac{1}{2}$ and $y = 1$ .
The solutions to the systems of equations are:
For system 7: $x = 7, y = 13$
For system 8: $x = \frac{1}{2}, y = 1$

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Question Solve the system of linear equations to find the common solution.x+y=20x+y=20x+y=20 and y=2x−1y=2x-1y=2x−1 y=2xy=2xy=2x and y=−6x+4y=-6x+4y=−6x+4

Studdy Solution

STEP 1

Assumptions1. We are given two systems of linear equations.2. Each system has two equations with two variables, xxx and yyy.3. We need to find the common solution for each system, which means finding the values of xxx and yyy that satisfy both equations in the system.

STEP 2

Solve the first system of equations:x+y=20y=2x−1 \begin{array}{l} x + y = 20 \\ y = 2x - 1 \end{array} x+y=20y=2x−1​

STEP 3

Substitute the expression for yyy from the second equation into the first equation.x+(2x−1)=20 x + (2x - 1) = 20 x+(2x−1)=20

STEP 4

Combine like terms in the equation.3x−1=20 3x - 1 = 20 3x−1=20

STEP 5

Add 1 to both sides of the equation to isolate the term with xxx.3x=21 3x = 21 3x=21

STEP 6

Divide both sides by 3 to solve for xxx.x=7 x = 7 x=7

STEP 7

Now that we have the value of xxx, we can substitute it back into the second equation to solve for yyy.y=2(7)−1 y = 2(7) - 1 y=2(7)−1

STEP 8

Calculate the value of yyy.y=14−1 y = 14 - 1 y=14−1y=13 y = 13 y=13

STEP 9

The common solution for the first system of equations is x=7x = 7x=7 and y=13y = 13y=13.

STEP 10

Now, solve the second system of equations:y=2xy=−6x+4 \begin{array}{l} y = 2x \\ y = -6x + 4 \end{array} y=2xy=−6x+4​

STEP 11

Since both equations equal yyy, we can set them equal to each other to find xxx.2x=−6x+4 2x = -6x + 4 2x=−6x+4

STEP 12

Add 6x6x6x to both sides of the equation to get all xxx terms on one side.8x=4 8x = 4 8x=4

STEP 13

Divide both sides by 8 to solve for xxx.x=48 x = \frac{4}{8} x=84​

STEP 14

Simplify the fraction to find the value of xxx.x=12 x = \frac{1}{2} x=21​

STEP 15

Now that we have the value of xxx, we can substitute it back into either of the original equations to solve for yyy. We'll use the first equation.y=2(12) y = 2 \left(\frac{1}{2}\right) y=2(21​)

STEP 16

Calculate the value of yyy.y=1 y = 1 y=1

STEP 17

The common solution for the second system of equations is x=12x = \frac{1}{2}x=21​ and y=1y = 1y=1.The solutions to the systems of equations are:For system 7: x=7,y=13 x = 7, y = 13 x=7,y=13For system 8: x=12,y=1 x = \frac{1}{2}, y = 1 x=21​,y=1

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Question Solve the system of linear equations to find the common solution.
$x+y=20$ and $y=2x-1$ $y=2x$ and $y=-6x+4$

Assumptions
1. We are given two systems of linear equations.
2. Each system has two equations with two variables, $x$ and $y$ .
3. We need to find the common solution for each system, which means finding the values of $x$ and $y$ that satisfy both equations in the system.

Solve the first system of equations:
$\begin{array}{l} x + y = 20 \\ y = 2x - 1 \end{array}$

Substitute the expression for $y$ from the second equation into the first equation.
$x + (2x - 1) = 20$

Combine like terms in the equation.
$3x - 1 = 20$

Add 1 to both sides of the equation to isolate the term with $x$ .
$3x = 21$

Divide both sides by 3 to solve for $x$ .
$x = 7$

Now that we have the value of $x$ , we can substitute it back into the second equation to solve for $y$ .
$y = 2(7) - 1$

Calculate the value of $y$ .
$y = 14 - 1$
$y = 13$

The common solution for the first system of equations is $x = 7$ and $y = 13$ .

Now, solve the second system of equations:
$\begin{array}{l} y = 2x \\ y = -6x + 4 \end{array}$

Since both equations equal $y$ , we can set them equal to each other to find $x$ .
$2x = -6x + 4$

Add $6x$ to both sides of the equation to get all $x$ terms on one side.
$8x = 4$

Divide both sides by 8 to solve for $x$ .
$x = \frac{4}{8}$

Simplify the fraction to find the value of $x$ .
$x = \frac{1}{2}$

Now that we have the value of $x$ , we can substitute it back into either of the original equations to solve for $y$ . We'll use the first equation.
$y = 2 \left(\frac{1}{2}\right)$

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

The common solution for the second system of equations is $x = \frac{1}{2}$ and $y = 1$ .
The solutions to the systems of equations are:
For system 7: $x = 7, y = 13$
For system 8: $x = \frac{1}{2}, y = 1$