Question

find all solutions of the system of equations algebraically. write your solutions as coordinate points.
$y = x^2 - 4x + 6$
$x + y = 6$
answer
two solutions
and
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Explanation:

Step1: Solve the linear equation for y

From \( x + y = 6 \), we can express \( y \) as \( y = 6 - x \).

Step2: Substitute y into the quadratic equation

Substitute \( y = 6 - x \) into \( y = x^2 - 4x + 6 \). So we get \( 6 - x = x^2 - 4x + 6 \).

Step3: Simplify the equation

Subtract 6 from both sides: \( -x = x^2 - 4x \). Then move all terms to one side: \( x^2 - 4x + x = 0 \), which simplifies to \( x^2 - 3x = 0 \).

Step4: Factor the quadratic equation

Factor out \( x \): \( x(x - 3) = 0 \).

Step5: Solve for x

Set each factor equal to zero: \( x = 0 \) or \( x - 3 = 0 \), so \( x = 0 \) or \( x = 3 \).

Step6: Find the corresponding y values

For \( x = 0 \), substitute into \( y = 6 - x \), we get \( y = 6 - 0 = 6 \). For \( x = 3 \), substitute into \( y = 6 - x \), we get \( y = 6 - 3 = 3 \).

Answer:

\((0, 6)\) and \((3, 3)\)