We can check our answers by substituting the points into both equations. If they both produce a true statement, our solutions are correct. Let's start by checking

(2, - 5) .

We will substitute

2

and

- 5

for

x

and

y,

respectively, in Equation (I) and Equation (II).

{y = x^{2} - 3 x - 3 y = - 2 x^{2} - x + 5

$(I), (II):$ $x = 2$ , $y = - 5$

{- 5 = ? (2)^{2} - 3 (2) - 3 - 5 = ? - 2 (2)^{2} - 2 + 5

▼

Simplify

$(I), (II):$ Calculate power

{- 5 = ? 4 - 3 (2) - 3 - 5 = ? - 2 (4) - 2 + 5

$(I), (II):$ $(- a) b = - a b$

{- 5 = ? 4 - 6 - 3 - 5 = ? - 8 - 2 + 5

$(I), (II):$ Add and subtract terms

{- 5 = - 5 ✓ - 5 = - 5 ✓

Since both equations produce true statements, the solution

(2, - 5)

is correct. Let's now check

(- \frac{4}{3}, \frac{2 5}{9}) .

{y = x^{2} - 3 x - 3 y = - 2 x^{2} - x + 5

$(I), (II):$ $x = - \frac{4}{3}$ , $y = \frac{2 5}{9}$

{\frac{2 5}{9} = ? (- \frac{4}{3})^{2} - 3 (- \frac{4}{3}) - 3 \frac{2 5}{9} = ? - 2 (- \frac{4}{3})^{2} - (- \frac{4}{3}) + 5

▼

Simplify

$(I), (II):$ Calculate power

{\frac{2 5}{9} = ? \frac{1 6}{9} - 3 (- \frac{4}{3}) - 3 \frac{2 5}{9} = ? - 2 (\frac{1 6}{9}) - (- \frac{4}{3}) + 5

MultNegNegOnePar

$(I):$ $- a (- b) = a \cdot b$

{\frac{2 5}{9} = ? \frac{1 6}{9} + 3 (\frac{4}{3}) - 3 \frac{2 5}{9} = ? - 2 (\frac{1 6}{9}) - (- \frac{4}{3}) + 5

DenomMultFracToNumber

$(I):$ $3 \cdot \frac{a}{3} = a$

{\frac{2 5}{9} = ? \frac{1 6}{9} + 4 - 3 \frac{2 5}{9} = ? - 2 (\frac{1 6}{9}) - (- \frac{4}{3}) + 5

SubTerm

$(I):$ Subtract term

{\frac{2 5}{9} = ? \frac{1 6}{9} + 1 \frac{2 5}{9} = ? - 2 (\frac{1 6}{9}) - (- \frac{4}{3}) + 5

MoveLeftFacToNum

$(II):$ $a \cdot \frac{b}{c} = \frac{a \cdot b}{c}$

{\frac{2 5}{9} = ? \frac{1 6}{9} + 1 \frac{2 5}{9} = ? - \frac{3 2}{9} - (- \frac{4}{3}) + 5

SubNeg

$(II):$ $a - (- b) = a + b$

{\frac{2 5}{9} = ? \frac{1 6}{9} + 1 \frac{2 5}{9} = ? - \frac{3 2}{9} + \frac{4}{3} + 5

ExpandFrac

$(II):$ $\frac{a}{b} = \frac{a \cdot 3}{b \cdot 3}$

{\frac{2 5}{9} = ? \frac{1 6}{9} + 1 \frac{2 5}{9} = ? - \frac{3 2}{9} + \frac{1 2}{9} + 5

$(I), (II):$ $a = \frac{9 \cdot a}{9}$

{\frac{2 5}{9} = ? \frac{1 6}{9} + \frac{9}{9} \frac{2 5}{9} = ? - \frac{3 2}{9} + \frac{1 2}{9} + \frac{4 5}{9}

$(I), (II):$ Add and subtract fractions

{\frac{2 5}{9} = \frac{2 5}{9} ✓ \frac{2 5}{9} = \frac{2 5}{9} ✓

Since both equations produce true statements again, the solution

(- \frac{4}{3}, \frac{2 5}{9})

is also correct.

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