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Consider the following system of equations:

-x+y=5
y=(x-3)^2+2
Is (1, 6) a solution to the system of equations?

3 answers

To determine if the point \((1, 6)\) is a solution to the given system of equations, we need to substitute \(x = 1\) and \(y = 6\) into both equations.

  1. For the first equation: \[ -x + y = 5 \] Substituting \(x = 1\) and \(y = 6\): \[ -1 + 6 = 5 \] This simplifies to: \[ 5 = 5 \] This equation is satisfied.

  2. For the second equation: \[ y = (x - 3)^2 + 2 \] Substituting \(x = 1\) and \(y = 6\): \[ 6 = (1 - 3)^2 + 2 \] Calculate \((1 - 3)^2\): \[ (1 - 3)^2 = (-2)^2 = 4 \] Now substitute back into the equation: \[ 6 = 4 + 2 \] This simplifies to: \[ 6 = 6 \] This equation is also satisfied.

Since both equations are satisfied with the point \((1, 6)\), we conclude that \((1, 6)\) is indeed a solution to the system of equations.

Solve the system of equations using substitution.

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

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

Step 1: Solve the first equation for \(y\).

From the first equation: \[ y = x + 5 \]

Step 2: Substitute \(y\) into the second equation.

Now we can substitute \(y = x + 5\) into the second equation: \[ x + 5 = (x - 3)^2 + 2 \]

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

We will first expand \((x - 3)^2\): \[ x + 5 = (x^2 - 6x + 9) + 2 \] \[ x + 5 = x^2 - 6x + 11 \]

Now, rearranging the equation gives: \[ 0 = x^2 - 6x + 11 - x - 5 \] \[ 0 = x^2 - 7x + 6 \]

Step 4: Factor the quadratic.

This can be factored as: \[ 0 = (x - 1)(x - 6) \]

Step 5: Solve for \(x\).

Setting each factor to zero gives us: \[ x - 1 = 0 \quad \Rightarrow \quad x = 1 \] \[ x - 6 = 0 \quad \Rightarrow \quad x = 6 \]

Step 6: Find corresponding \(y\) values.

Now we will find the corresponding \(y\) values using the equation \(y = x + 5\).

  1. If \(x = 1\): \[ y = 1 + 5 = 6 \] So one solution is \((1, 6)\).

  2. If \(x = 6\): \[ y = 6 + 5 = 11 \] So the other solution is \((6, 11)\).

Final Solutions: The solutions to the system of equations are: \[ (1, 6) \quad \text{and} \quad (6, 11) \]