Solve the system of two equations using substitution. y=2+3x 4x−3y=−16 (1 point) Responses (2, 8) left parenthesis 2 comma 8 right parenthesis (−2, 4) left parenthesis negative 2 comma 4 right parenthesis (−2, −4) left parenthesis negative 2 comma negative 4 right parenthesis (2, 10)

Answer 1

To solve the system of equations using substitution, we have:

1. \( y = 2 + 3x \)
2. \( 4x - 3y = -16 \)

We can substitute the expression for \( y \) from the first equation into the second equation:

Substituting \( y \) into the second equation:

\[ 4x - 3(2 + 3x) = -16 \]

Now, distribute the -3:

\[ 4x - 6 - 9x = -16 \]

Combine like terms:

\[ -5x - 6 = -16 \]

Now, add 6 to both sides:

\[ -5x = -10 \]

Dividing both sides by -5 gives:

\[ x = 2 \]

Now that we have \( x \), we can substitute it back into the first equation to find \( y \):

\[ y = 2 + 3(2) = 2 + 6 = 8 \]

So the solution to the system of equations is:

\[ (x, y) = (2, 8) \]

Thus, the answer is:

\((2, 8)\)

Answer 2

Solve the system of equations. 2x+6y=−18 x=5y−1 (1 point) Responses (−7.25, −1.25) left parenthesis negative 7.25 comma negative 1.25 right parenthesis (5.25, 1.25) left parenthesis 5.25 comma 1.25 right parenthesis (−6, −1) left parenthesis negative 6 comma negative 1 right parenthesis (4, 1)

Answer 3

To solve the system of equations:

1. \( 2x + 6y = -18 \)
2. \( x = 5y - 1 \)

We can use substitution. From the second equation, we have \( x \) expressed in terms of \( y \):

\[ x = 5y - 1 \]

Now, we can substitute this expression for \( x \) into the first equation:

\[ 2(5y - 1) + 6y = -18 \]

Distributing the \( 2 \):

\[ 10y - 2 + 6y = -18 \]

Combine like terms:

\[ 16y - 2 = -18 \]

Now, add \( 2 \) to both sides:

\[ 16y = -16 \]

Dividing both sides by \( 16 \):

\[ y = -1 \]

Now, substitute \( y = -1 \) back into the equation for \( x \):

\[ x = 5(-1) - 1 = -5 - 1 = -6 \]

Thus, the solution to the system of equations is:

\[ (x, y) = (-6, -1) \]

So, the answer is:

\((-6, -1)\)

Answer 4

Solve the system of equations. 3x+y=2 7x−4y=30 (1 point) Responses (2, −4) left parenthesis 2 comma negative 4 right parenthesis (12, 12) left parenthesis start fraction 1 over 2 end fraction comma start fraction 1 over 2 end fraction right parenthesis (−2, 8) left parenthesis negative 2 comma 8 right parenthesis (2, −2)

Answer 5

To solve the system of equations:

1. \( 3x + y = 2 \)
2. \( 7x - 4y = 30 \)

We can use substitution or elimination. Here, let's use substitution by solving the first equation for \( y \):

\[ y = 2 - 3x \]

Now, we can substitute this expression for \( y \) into the second equation:

\[ 7x - 4(2 - 3x) = 30 \]

Distributing the \(-4\):

\[ 7x - 8 + 12x = 30 \]

Combine like terms:

\[ 19x - 8 = 30 \]

Now, add \( 8 \) to both sides:

\[ 19x = 38 \]

Dividing both sides by \( 19 \):

\[ x = 2 \]

Now that we have \( x \), we can substitute it back into the expression for \( y \):

\[ y = 2 - 3(2) = 2 - 6 = -4 \]

So the solution to the system of equations is:

\[ (x, y) = (2, -4) \]

Thus, the answer is:

\((2, -4)\)

Answer 6

Solve the system of equations. −4x+6y=38 x−5y=−20 (1 point) Responses (−35, −3) left parenthesis negative 35 comma negative 3 right parenthesis (−5, 3) left parenthesis negative 5 comma 3 right parenthesis (5, 3) left parenthesis 5 comma 3 right parenthesis (35, −3)

Answer 7

To solve the system of equations:

1. \( -4x + 6y = 38 \)
2. \( x - 5y = -20 \)

We can use the substitution or elimination method. Here, I'll use the elimination method for this example.

Step 1: Rearrange the second equation

First, we can isolate \( x \) in the second equation:

\[ x = 5y - 20 \]

Step 2: Substitute into the first equation

Now we'll substitute \( x \) from the second equation into the first equation:

\[ -4(5y - 20) + 6y = 38 \]

Step 3: Distribute and simplify

Distributing \(-4\):

\[ -20y + 80 + 6y = 38 \]

Combine the \( y \) terms:

\[ -14y + 80 = 38 \]

Step 4: Solve for \( y \)

Now, subtract \( 80 \) from both sides:

\[ -14y = 38 - 80 \] \[ -14y = -42 \]

Now, divide by \(-14\):

\[ y = 3 \]

Step 5: Substitute back to find \( x \)

Now that we have \( y \), we can substitute \( y = 3 \) back into the equation for \( x \):

\[ x = 5(3) - 20 \] \[ x = 15 - 20 \] \[ x = -5 \]

Solution

Thus, the solution to the system of equations is:

\[ (x, y) = (-5, 3) \]

So, the answer is:

\((-5, 3)\)

Answer 8

How many solution(s) does the system of equations have? −5x+y=−18 10x−2y=36 (1 point) Responses two two infinite infinite one one none

Answer 9

To determine how many solutions the system of equations has, we need to analyze the equations:

1. \( -5x + y = -18 \)
2. \( 10x - 2y = 36 \)

Let's first manipulate these equations to see if they are equivalent or consistent.

Step 1: Rewrite the equations

We can rearrange the first equation:

\[ y = 5x - 18 \]

For the second equation, we can rewrite it in a similar form. We can isolate \( y \):

\[ 10x - 2y = 36 \] \[ -2y = -10x + 36 \] \[ y = 5x - 18 \]

Step 2: Compare the two equations

Both equations can be rewritten as:

\[ y = 5x - 18 \]

Since both equations are identical, they represent the same line. This means that every point on this line is a solution to the system of equations.

Conclusion

The system of equations has infinitely many solutions.

Therefore, the answer is infinite.

Answer 10

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Answer 11

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