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Solve the system by substitution.

minus, 5, x, plus, 3, y, equals, 2
−5x+3y=
2
y, equals, 2, x
y=
2x

11 answers

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

  1. \(-5x + 3y = 2\) (Equation 1)
  2. \(y = 2x\) (Equation 2)

Step 1: Substitute Equation 2 into Equation 1

We can substitute \(y\) from Equation 2 into Equation 1:

\[ -5x + 3(2x) = 2 \]

Step 2: Simplify the equation

Now, simplify the equation:

\[ -5x + 6x = 2 \]

This simplifies to:

\[ x = 2 \]

Step 3: Solve for \(y\)

Now that we have \(x\), we can use Equation 2 to find \(y\):

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

Final Solution

The solution to the system of equations is:

\[ x = 2, \quad y = 4 \]

In coordinate form, the solution is \((2, 4)\).

just the answer pls

The solution is \((2, 4)\).

Solve the system by substitution.
3, x, minus, 6, equals, y
3x−6=
y
minus, 6, x, plus, y, equals, minus, 9
−6x+y=
−9

The solution to the system of equations is \((x, y) = (1, 3)\).

rong

I apologize for the error. Let's solve the system of equations again step-by-step.

We have the equations:

  1. \( y = 3x - 6 \) (Equation 1)
  2. \( -6x + y = -9 \) (Equation 2)

Step 1: Substitute Equation 1 into Equation 2

Substituting \( y \) from Equation 1 into Equation 2:

\[ -6x + (3x - 6) = -9 \]

Step 2: Simplify the equation

Combine like terms:

\[ -6x + 3x - 6 = -9 \]

This simplifies to:

\[ -3x - 6 = -9 \]

Step 3: Solve for \( x \)

Add 6 to both sides:

\[ -3x = -3 \]

Now, divide by -3:

\[ x = 1 \]

Step 4: Solve for \( y \)

Now, substitute \( x = 1 \) back into Equation 1 to find \( y \):

\[ y = 3(1) - 6 = 3 - 6 = -3 \]

Final Solution

The correct solution to the system of equations is:

\((x, y) = (1, -3)\).

Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution.
minus, x, plus, 6, y, equals, 3
−x+6y=
3
3, x, minus, 18, y, equals, minus, 10
3x−18y=
−10
Answer
Multiple Choice Answers
Infinitely Many SolutionsInfinitely Many Solutions
One SolutionOne Solution
No SolutionsNo Solutions

To determine the solution type of the given system of equations, we will analyze the two equations.

  1. \(-x + 6y = 3\) (Equation 1)
  2. \(3x - 18y = -10\) (Equation 2)

Step 1: Modify Equation 1 for comparison

Let's multiply Equation 1 by 3 to compare the two equations:

\[ 3(-x + 6y) = 3(3) \]

This simplifies to:

\[ -3x + 18y = 9 \quad \text{(Equation 1 modified)} \]

Step 2: Rewrite Equation 2

Now, compare this modified equation to Equation 2:

\[ 3x - 18y = -10 \]

Step 3: Check for consistency

Now we can see:

  • Equation 1 modified: \(-3x + 18y = 9\)
  • Equation 2: \(3x - 18y = -10\)

To clarify, we can rearrange each equation to compare their slopes and intercepts:

From the modified Equation 1:

\[ 3x - 18y = -9 \quad (multiply by -1) \]

Now we have:

  • Modified Equation 1: \(3x - 18y = -9\)
  • Equation 2: \(3x - 18y = -10\)

Observations

The left sides of both equations are identical (\(3x - 18y\)), but the right sides do not equal each other \((-9 \neq -10)\). This indicates that the lines represented by these equations are parallel and will never intersect, meaning there is no solution.

Conclusion

Thus, the system of equations has no solutions.

Solve the following system of equations algebraically:
y, equals, x, squared, minus, 5, x, plus, 4
y=
x
2
−5x+4
y, equals, x, plus, 4
y=
x+4

To solve the system of equations algebraically, we have the following:

  1. \(y = x^2 - 5x + 4\) (Equation 1)
  2. \(y = x + 4\) (Equation 2)

Step 1: Set the two equations equal to each other

Since both equations equal \(y\), we can set them equal to each other:

\[ x^2 - 5x + 4 = x + 4 \]

Step 2: Rearrange the equation

Move all terms to one side:

\[ x^2 - 5x + 4 - x - 4 = 0 \]

This simplifies to:

\[ x^2 - 6x = 0 \]

Step 3: Factor the equation

Now, factor the equation:

\[ x(x - 6) = 0 \]

Step 4: Solve for \(x\)

Set each factor equal to zero:

  1. \(x = 0\)
  2. \(x - 6 = 0 \Rightarrow x = 6\)

Step 5: Solve for \(y\)

Now, substitute \(x = 0\) and \(x = 6\) back into Equation 2 to find the corresponding \(y\) values.

  1. For \(x = 0\):

\[ y = 0 + 4 = 4 \]

So, one solution is \((0, 4)\).

  1. For \(x = 6\):

\[ y = 6 + 4 = 10 \]

So, the other solution is \((6, 10)\).

Final Solution

The solutions to the system of equations are:

\((0, 4)\) and \((6, 10)\).