Solve the system of two equations using substitution.

Question

y = 2 + 3x

4x − 3y = −16

option:

(−2, 4)

(2, 10)

(−2, −4)

(2, 8)

Answer 1

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

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

Since the first equation already gives us \( y \) in terms of \( x \), we can substitute this expression for \( y \) into the second equation.

Substituting \( y \) in the second equation:

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

Now, distribute the \(-3\) to the terms inside the parentheses:

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

Combine like terms:

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

This simplifies to:

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

Next, add 6 to both sides:

\[ -5x = -10 \]

Now, divide both sides by -5:

\[ x = 2 \]

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

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

\[ y = 2 + 6 = 8 \]

Thus, the solution to the system of equations is:

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

From the options provided, the correct answer is:

(2, 8).

Answer 2

Solve the system of equations.
2x+6y=−18
x=5y−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 \( y \). We'll substitute \( x = 5y - 1 \) into the first equation.

Substituting \( x \) into the first equation:

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

Now, distribute the 2:

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

Combine like terms:

\[ 16y - 2 = -18 \]

Next, add 2 to both sides:

\[ 16y = -16 \]

Now, divide both sides by 16:

\[ y = -1 \]

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

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

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

So the solution to the system of equations is:

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

Thus, the solution is \( (x, y) = (-6, -1) \).

Answer 4

Solve the system of equations.
3x+y=2
7x−4y=30

Answer 5

To solve the system of equations:

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

we can use substitution or elimination. Here, I'll use substitution.

First, we can solve the first equation for \( y \):

\[ y = 2 - 3x \]

Now that we have \( y \) in terms of \( x \), we can substitute this expression into the second equation:

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

Now, distribute \(-4\):

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

Combine like terms:

\[ 19x - 8 = 30 \]

Next, add 8 to both sides:

\[ 19x = 38 \]

Now, divide both sides by 19:

\[ x = 2 \]

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

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

\[ y = 2 - 6 = -4 \]

Thus, the solution to the system of equations is:

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

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

Answer 6

Solve the system of equations.
−4x+6y=38
x−5y=−20

Answer 7

To solve the system of equations:

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

we will use the method of substitution or elimination. I'll use the substitution method for this problem.

First, let's solve the second equation \(x - 5y = -20\) for \(x\):

\[ x = 5y - 20 \]

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

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

Now distribute \(-4\):

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

Combine like terms:

\[ -20y + 6y + 80 = 38 \] \[ -14y + 80 = 38 \]

Next, subtract 80 from both sides:

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

Now, divide by \(-14\):

\[ y = \frac{-42}{-14} = 3 \]

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

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

Thus, the solution to the system of equations is:

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

So the final answer is \((-5, 3)\).

Answer 8

How many solution(s) does the system of equations have?
−5x+y=−18
10x−2y=36

Answer 9

To determine how many solutions the given system of equations has, we can examine the equations:

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

First, let's write both equations in standard form.

Step 1: Rewrite Equations

Rearranging the first equation for \(y\):

\[ y = 5x - 18 \]

The second equation is already in standard form but let's rewrite it for clarity:

\[ 10x - 2y = 36 \]

Step 2: Determine Consistency

Now, we can observe whether the two lines represented by these equations are consistent (having one solution), inconsistent (having no solutions), or dependent (having infinitely many solutions).

To check consistency, we can try to manipulate both equations or find the slopes of the lines.

Step 3: Manipulating the Second Equation

We can write the second equation in terms of \(y\):

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

Dividing everything by \(-2\) gives:

\[ y = 5x - 18 \]

Step 4: Compare the Lines

Now we see that both equations simplify to:

1. From the first equation: \(y = 5x - 18\)
2. From the second equation: \(y = 5x - 18\)

Both equations are identical.

Conclusion

Since both equations represent the same line, the system has infinitely many solutions. Any point on the line is a solution to both equations.

Thus, the answer is infinitely many solutions.