Use the substitution method to solve the system y=−3 , y=2x+1 .

Answer 1

To solve the system of equations using the substitution method, we start with the given equations:

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

Since the first equation gives us a direct value for \( y \), we can substitute this value into the second equation.

Substituting \( y = -3 \) into the second equation:

\[ -3 = 2x + 1 \]

Next, we will solve for \( x \).

1. Subtract 1 from both sides:

\[ -3 - 1 = 2x \]

\[ -4 = 2x \]

2. Now, divide both sides by 2:

\[ x = -2 \]

Now that we have found \( x = -2 \), we can substitute this value back into the first equation to find \( y \):

\[ y = -3 \]

So the solution to the system of equations is:

\[ (x, y) = (-2, -3) \]

Thus, the solution is \( (-2, -3) \).

Answer 2

Use the substitution method to solve the system y=−2x+1
, −4x=−2y+10

Answer 3

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

1. \( y = -2x + 1 \)
2. \( -4x = -2y + 10 \)

We’ll start by substituting the expression for \( y \) from the first equation into the second equation.

Step 1: Substitute \( y \) in the second equation

Substituting \( y = -2x + 1 \) into the second equation:

\[ -4x = -2(-2x + 1) + 10 \]

Step 2: Simplify the equation

Now we will simplify the equation:

\[ -4x = -2(-2x) + (-2 \cdot 1) + 10 \] \[ -4x = 4x - 2 + 10 \] \[ -4x = 4x + 8 \]

Step 3: Solve for \( x \)

Now, let's get all terms involving \( x \) on one side:

\[ -4x - 4x = 8 \] \[ -8x = 8 \]

Now, divide both sides by -8:

\[ x = -1 \]

Step 4: Substitute \( x \) back to find \( y \)

Now that we have \( x = -1 \), we can substitute this back into the first equation to find \( y \):

\[ y = -2(-1) + 1 \] \[ y = 2 + 1 \] \[ y = 3 \]

Step 5: Write the solution

The solution to the system of equations is:

\[ (x, y) = (-1, 3) \]

Thus, the solution is \( (-1, 3) \).

Answer 4

Write an equation equivalent to 2x+3y=5
by multiplying the equation by 4.

Answer 5

To create an equivalent equation to \( 2x + 3y = 5 \) by multiplying the entire equation by 4, we proceed as follows:

Start with the original equation:

\[ 2x + 3y = 5 \]

Now, multiply each term in the equation by 4:

\[ 4(2x) + 4(3y) = 4(5) \]

This simplifies to:

\[ 8x + 12y = 20 \]

Thus, the equation equivalent to \( 2x + 3y = 5 \) by multiplying by 4 is:

\[ 8x + 12y = 20 \]

Answer 6

What operation needs to be performed to go from the equation 2x+7y=14
to the equivalent equation 4x+14y=28
.The equation 2x+7y=14
should be multiplied by ( )to produce the equivalent equation 4x+14y=28
.

Answer 7

To find what operation needs to be performed to go from the equation \( 2x + 7y = 14 \) to the equation \( 4x + 14y = 28 \), we can analyze the transformation between the two equations.

Starting with the original equation:

\[ 2x + 7y = 14 \]

Now let's look at the target equation:

\[ 4x + 14y = 28 \]

Notice that each term in \( 4x + 14y = 28 \) can be related to the corresponding term in \( 2x + 7y = 14 \) by multiplying by 2:

• \( 2x \) multiplied by 2 gives \( 4x \)
• \( 7y \) multiplied by 2 gives \( 14y \)
• \( 14 \) multiplied by 2 gives \( 28 \)

Thus, to go from \( 2x + 7y = 14 \) to \( 4x + 14y = 28 \), we multiply the entire equation by 2.

Therefore, the equation \( 2x + 7y = 14 \) should be multiplied by 2 to produce the equivalent equation \( 4x + 14y = 28 \).

Answer 8

Solve the following system of equations using the elimination method.

2x−3y=14
, 4x+3y=10

( )( )

Answer 9

To solve the system of equations using the elimination method, we have the following equations:

1. \( 2x - 3y = 14 \) (Equation 1)
2. \( 4x + 3y = 10 \) (Equation 2)

Step 1: Align the Equations

We'll rewrite the equations for clarity:

\[ 2x - 3y = 14 \quad (1) \] \[ 4x + 3y = 10 \quad (2) \]

Step 2: Eliminate \( y \)

To eliminate \( y \), we can add the two equations. However, to do that effectively, we'll first multiply Equation 1 by \( 1 \) (which keeps it the same) and Equation 2 by \( 1 \) as well, so we can add them directly. (There's no need to change anything in this case since adding the same coefficients of \( y \) will allow cancellation.)

Now we can just add the two equations:

\[ (2x - 3y) + (4x + 3y) = 14 + 10 \]

Step 3: Combine Like Terms

Combining the left-hand side:

\[ 2x + 4x + (-3y + 3y) = 14 + 10 \] \[ 6x + 0y = 24 \]

This simplifies to:

\[ 6x = 24 \]

Step 4: Solve for \( x \)

Now, we can solve for \( x \):

\[ x = \frac{24}{6} \] \[ x = 4 \]

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

Now that we have \( x = 4 \), we substitute this value back into either of the original equations to find \( y \). We'll use Equation 1:

\[ 2(4) - 3y = 14 \]

This simplifies to:

\[ 8 - 3y = 14 \]

Now, isolate \( y \):

1. Subtract \( 8 \) from both sides:

\[ -3y = 14 - 8 \] \[ -3y = 6 \]

2. Now, divide by \( -3 \):

\[ y = \frac{6}{-3} \] \[ y = -2 \]

Final Solution

The solution to the system of equations is:

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

So, the answer is \( (4, -2) \).

Answer 10

Use the substitution method to solve the system 2x+3y=4, −4x−6y=−8.(1 point)
Responses

(2−1)
left parenthesis 2 comma negative 1 right parenthesis

(−1,2)
left parenthesis negative 1 comma 2 right parenthesis

no solution
no solution

infinite solutions

Answer 11

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

1. \( 2x + 3y = 4 \) (Equation 1)
2. \( -4x - 6y = -8 \) (Equation 2)

Step 1: Solve one of the equations for one variable

Let's solve Equation 1 for \( y \):

\[ 2x + 3y = 4 \]

Subtract \( 2x \) from both sides:

\[ 3y = 4 - 2x \]

Now, divide by 3:

\[ y = \frac{4 - 2x}{3} \]

Step 2: Substitute into the second equation

Now, substitute this expression for \( y \) into Equation 2:

\[ -4x - 6y = -8 \]

Substituting \( y \):

\[ -4x - 6\left(\frac{4 - 2x}{3}\right) = -8 \]

Step 3: Simplify and solve for \( x \)

Distributing the \(-6\):

\[ -4x - \frac{24 - 12x}{3} = -8 \]

To eliminate the fraction, multiply the entire equation by 3:

\[ 3(-4x) - (24 - 12x) = -24 \]

This simplifies to:

\[ -12x - 24 + 12x = -24 \]

Now, the \( -12x \) and \( +12x \) cancel out:

\[ -24 = -24 \]

Conclusion

Since we ended up with the true statement \( -24 = -24 \), this indicates that the two equations are not independent; they represent the same line.

Therefore, there are infinite solutions to this system.

The conclusion is:

Infinite solutions