Solve the System of Equations y=-x+3 y=2x^2-3x+4

$y = - x + 3$ $y = 2 x^{2} - 3 x + 4$

Step 1

Eliminate the equal sides of each equation and combine.

$- x + 3 = 2 x^{2} - 3 x + 4$

Step 2

Solve $- x + 3 = 2 x^{2} - 3 x + 4$ for $x$ .

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$2 x^{2} - 3 x + 4 = - x + 3$

Move all terms containing $x$ to the left side of the equation.

Add $x$ to both sides of the equation.

$2 x^{2} - 3 x + 4 + x = 3$

Add $- 3 x$ and $x$ .

$2 x^{2} - 2 x + 4 = 3$

Move all terms to the left side of the equation and simplify.

Subtract $3$ from both sides of the equation.

$2 x^{2} - 2 x + 4 - 3 = 0$

Subtract $3$ from $4$ .

$2 x^{2} - 2 x + 1 = 0$

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a} + y = 2 x^{2} - 3 x + 4$

Substitute the values $a = 2$ , $b = - 2$ , and $c = 1$ into the quadratic formula and solve for $x$ .

$\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (2 \cdot 1)}}{2 \cdot 2} + y = 2 x^{2} - 3 x + 4$

Simplify.

Simplify the numerator.

Raise $- 2$ to the power of $2$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 2 \cdot 1}}{2 \cdot 2}$

Multiply $- 4 \cdot 2 \cdot 1$ .

Multiply $- 4$ by $2$ .

$x = \frac{2 \pm \sqrt{4 - 8 \cdot 1}}{2 \cdot 2}$

Multiply $- 8$ by $1$ .

$x = \frac{2 \pm \sqrt{4 - 8}}{2 \cdot 2}$

Subtract $8$ from $4$ .

$x = \frac{2 \pm \sqrt{- 4}}{2 \cdot 2}$

Rewrite $- 4$ as $- 1 (4)$ .

$x = \frac{2 \pm \sqrt{- 1 \cdot 4}}{2 \cdot 2}$

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$x = \frac{2 \pm \sqrt{- 1} \cdot \sqrt{4}}{2 \cdot 2}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{2 \pm i \cdot \sqrt{4}}{2 \cdot 2}$

Rewrite $4$ as $2^{2}$ .

$x = \frac{2 \pm i \cdot \sqrt{2^{2}}}{2 \cdot 2}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{2 \pm i \cdot 2}{2 \cdot 2}$

Move $2$ to the left of $i$ .

$x = \frac{2 \pm 2 i}{2 \cdot 2}$

Multiply $2$ by $2$ .

$x = \frac{2 \pm 2 i}{4}$

Simplify $\frac{2 \pm 2 i}{4}$ .

$x = \frac{1 \pm i}{2}$

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Simplify the numerator.

Raise $- 2$ to the power of $2$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 2 \cdot 1}}{2 \cdot 2}$

Multiply $- 4 \cdot 2 \cdot 1$ .

Multiply $- 4$ by $2$ .

$x = \frac{2 \pm \sqrt{4 - 8 \cdot 1}}{2 \cdot 2}$

Multiply $- 8$ by $1$ .

$x = \frac{2 \pm \sqrt{4 - 8}}{2 \cdot 2}$

Subtract $8$ from $4$ .

$x = \frac{2 \pm \sqrt{- 4}}{2 \cdot 2}$

Rewrite $- 4$ as $- 1 (4)$ .

$x = \frac{2 \pm \sqrt{- 1 \cdot 4}}{2 \cdot 2}$

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$x = \frac{2 \pm \sqrt{- 1} \cdot \sqrt{4}}{2 \cdot 2}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{2 \pm i \cdot \sqrt{4}}{2 \cdot 2}$

Rewrite $4$ as $2^{2}$ .

$x = \frac{2 \pm i \cdot \sqrt{2^{2}}}{2 \cdot 2}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{2 \pm i \cdot 2}{2 \cdot 2}$

Move $2$ to the left of $i$ .

$x = \frac{2 \pm 2 i}{2 \cdot 2}$

Multiply $2$ by $2$ .

$x = \frac{2 \pm 2 i}{4}$

Simplify $\frac{2 \pm 2 i}{4}$ .

$x = \frac{1 \pm i}{2}$

Change the $\pm$ to $+$ .

$x = \frac{1 + i}{2}$

Split the fraction $\frac{1 + i}{2}$ into two fractions.

$x = \frac{1}{2} + \frac{i}{2}$

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Simplify the numerator.

Raise $- 2$ to the power of $2$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 2 \cdot 1}}{2 \cdot 2}$

Multiply $- 4 \cdot 2 \cdot 1$ .

Multiply $- 4$ by $2$ .

$x = \frac{2 \pm \sqrt{4 - 8 \cdot 1}}{2 \cdot 2}$

Multiply $- 8$ by $1$ .

$x = \frac{2 \pm \sqrt{4 - 8}}{2 \cdot 2}$

Subtract $8$ from $4$ .

$x = \frac{2 \pm \sqrt{- 4}}{2 \cdot 2}$

Rewrite $- 4$ as $- 1 (4)$ .

$x = \frac{2 \pm \sqrt{- 1 \cdot 4}}{2 \cdot 2}$

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$x = \frac{2 \pm \sqrt{- 1} \cdot \sqrt{4}}{2 \cdot 2}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{2 \pm i \cdot \sqrt{4}}{2 \cdot 2}$

Rewrite $4$ as $2^{2}$ .

$x = \frac{2 \pm i \cdot \sqrt{2^{2}}}{2 \cdot 2}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{2 \pm i \cdot 2}{2 \cdot 2}$

Move $2$ to the left of $i$ .

$x = \frac{2 \pm 2 i}{2 \cdot 2}$

Multiply $2$ by $2$ .

$x = \frac{2 \pm 2 i}{4}$

Simplify $\frac{2 \pm 2 i}{4}$ .

$x = \frac{1 \pm i}{2}$

Change the $\pm$ to $-$ .

$x = \frac{1 - i}{2}$

Split the fraction $\frac{1 - i}{2}$ into two fractions.

$x = \frac{1}{2} + \frac{- i}{2}$

Move the negative in front of the fraction.

$x = \frac{1}{2} - \frac{i}{2}$

The final answer is the combination of both solutions.

$x = \frac{1}{2} + \frac{i}{2}, \frac{1}{2} - \frac{i}{2}$

Step 3

Evaluate $y$ when $x = \frac{1}{2} + \frac{i}{2}$ .

Substitute $\frac{1}{2} + \frac{i}{2}$ for $x$ .

$y = 2 {(\frac{1}{2} + \frac{i}{2})}^{2} - 3 (\frac{1}{2} + \frac{i}{2}) + 4$

Simplify $2 {(\frac{1}{2} + \frac{i}{2})}^{2} - 3 (\frac{1}{2} + \frac{i}{2}) + 4$ .

Simplify each term.

Rewrite ${(\frac{1}{2} + \frac{i}{2})}^{2}$ as $(\frac{1}{2} + \frac{i}{2}) (\frac{1}{2} + \frac{i}{2})$ .

$y = 2 ((\frac{1}{2} + \frac{i}{2}) (\frac{1}{2} + \frac{i}{2})) - 3 (\frac{1}{2} + \frac{i}{2}) + 4$

Expand $(\frac{1}{2} + \frac{i}{2}) (\frac{1}{2} + \frac{i}{2})$ using the FOIL Method.

Apply the distributive property.

$y = 2 (\frac{1}{2} (\frac{1}{2} + \frac{i}{2}) + \frac{i}{2} (\frac{1}{2} + \frac{i}{2})) - 3 (\frac{1}{2} + \frac{i}{2}) + 4$

Apply the distributive property.

$y = 2 (\frac{1}{2} \cdot \frac{1}{2} + \frac{1}{2} \cdot \frac{i}{2} + \frac{i}{2} (\frac{1}{2} + \frac{i}{2})) - 3 (\frac{1}{2} + \frac{i}{2}) + 4$

Apply the distributive property.

$y = 2 (\frac{1}{2} \cdot \frac{1}{2} + \frac{1}{2} \cdot \frac{i}{2} + \frac{i}{2} \cdot \frac{1}{2} + \frac{i}{2} \cdot \frac{i}{2}) - 3 (\frac{1}{2} + \frac{i}{2}) + 4$

Simplify and combine like terms.

Simplify each term.

Multiply $\frac{1}{2} \cdot \frac{1}{2}$ .

Multiply $\frac{1}{2}$ by $\frac{1}{2}$ .

$y = 2 (\frac{1}{2 \cdot 2} + \frac{1}{2} \cdot \frac{i}{2} + \frac{i}{2} \cdot \frac{1}{2} + \frac{i}{2} \cdot \frac{i}{2}) - 3 (\frac{1}{2} + \frac{i}{2}) + 4$

Multiply $2$ by $2$ .

$y = 2 (\frac{1}{4} + \frac{1}{2} \cdot \frac{i}{2} + \frac{i}{2} \cdot \frac{1}{2} + \frac{i}{2} \cdot \frac{i}{2}) - 3 (\frac{1}{2} + \frac{i}{2}) + 4$

Multiply $\frac{1}{2} \cdot \frac{i}{2}$ .

Multiply $\frac{1}{2}$ by $\frac{i}{2}$ .

$y = 2 (\frac{1}{4} + \frac{i}{2 \cdot 2} + \frac{i}{2} \cdot \frac{1}{2} + \frac{i}{2} \cdot \frac{i}{2}) - 3 (\frac{1}{2} + \frac{i}{2}) + 4$

Multiply $2$ by $2$ .

$y = 2 (\frac{1}{4} + \frac{i}{4} + \frac{i}{2} \cdot \frac{1}{2} + \frac{i}{2} \cdot \frac{i}{2}) - 3 (\frac{1}{2} + \frac{i}{2}) + 4$

Multiply $\frac{i}{2} \cdot \frac{1}{2}$ .

Multiply $\frac{i}{2}$ by $\frac{1}{2}$ .

$y = 2 (\frac{1}{4} + \frac{i}{4} + \frac{i}{2 \cdot 2} + \frac{i}{2} \cdot \frac{i}{2}) - 3 (\frac{1}{2} + \frac{i}{2}) + 4$

Multiply $2$ by $2$ .

$y = 2 (\frac{1}{4} + \frac{i}{4} + \frac{i}{4} + \frac{i}{2} \cdot \frac{i}{2}) - 3 (\frac{1}{2} + \frac{i}{2}) + 4$

Multiply $\frac{i}{2} \cdot \frac{i}{2}$ .

Multiply $\frac{i}{2}$ by $\frac{i}{2}$ .

$y = 2 (\frac{1}{4} + \frac{i}{4} + \frac{i}{4} + \frac{i i}{2 \cdot 2}) - 3 (\frac{1}{2} + \frac{i}{2}) + 4$

Raise $i$ to the power of $1$ .

$y = 2 (\frac{1}{4} + \frac{i}{4} + \frac{i}{4} + \frac{i^{1} i}{2 \cdot 2}) - 3 (\frac{1}{2} + \frac{i}{2}) + 4$

Raise $i$ to the power of $1$ .

$y = 2 (\frac{1}{4} + \frac{i}{4} + \frac{i}{4} + \frac{i^{1} i^{1}}{2 \cdot 2}) - 3 (\frac{1}{2} + \frac{i}{2}) + 4$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = 2 (\frac{1}{4} + \frac{i}{4} + \frac{i}{4} + \frac{i^{1 + 1}}{2 \cdot 2}) - 3 (\frac{1}{2} + \frac{i}{2}) + 4$

Add $1$ and $1$ .

$y = 2 (\frac{1}{4} + \frac{i}{4} + \frac{i}{4} + \frac{i^{2}}{2 \cdot 2}) - 3 (\frac{1}{2} + \frac{i}{2}) + 4$

Multiply $2$ by $2$ .

$y = 2 (\frac{1}{4} + \frac{i}{4} + \frac{i}{4} + \frac{i^{2}}{4}) - 3 (\frac{1}{2} + \frac{i}{2}) + 4$

Rewrite $i^{2}$ as $- 1$ .

$y = 2 (\frac{1}{4} + \frac{i}{4} + \frac{i}{4} + \frac{- 1}{4}) - 3 (\frac{1}{2} + \frac{i}{2}) + 4$

Move the negative in front of the fraction.

$y = 2 (\frac{1}{4} + \frac{i}{4} + \frac{i}{4} - \frac{1}{4}) - 3 (\frac{1}{2} + \frac{i}{2}) + 4$

Combine the numerators over the common denominator.

$y = 2 (\frac{1 - 1}{4} + \frac{i}{4} + \frac{i}{4}) - 3 (\frac{1}{2} + \frac{i}{2}) + 4$

Subtract $1$ from $1$ .

$y = 2 (\frac{0}{4} + \frac{i}{4} + \frac{i}{4}) - 3 (\frac{1}{2} + \frac{i}{2}) + 4$

Combine the numerators over the common denominator.

$y = 2 (\frac{0}{4} + \frac{2 i}{4}) - 3 (\frac{1}{2} + \frac{i}{2}) + 4$

Simplify each term.

Divide $0$ by $4$ .

$y = 2 (0 + \frac{2 i}{4}) - 3 (\frac{1}{2} + \frac{i}{2}) + 4$

Cancel the common factor of $2$ and $4$ .

Factor $2$ out of $2 i$ .

$y = 2 (0 + \frac{2 (i)}{4}) - 3 (\frac{1}{2} + \frac{i}{2}) + 4$

Cancel the common factors.

Factor $2$ out of $4$ .

$y = 2 (0 + \frac{2 i}{2 \cdot 2}) - 3 (\frac{1}{2} + \frac{i}{2}) + 4$

Cancel the common factor.

$y = 2 (0 + \frac{2 i}{2 \cdot 2}) - 3 (\frac{1}{2} + \frac{i}{2}) + 4$

Rewrite the expression.

$y = 2 (0 + \frac{i}{2}) - 3 (\frac{1}{2} + \frac{i}{2}) + 4$

Add $0$ and $\frac{i}{2}$ .

$y = 2 \frac{i}{2} - 3 (\frac{1}{2} + \frac{i}{2}) + 4$

Cancel the common factor of $2$ .

Cancel the common factor.

$y = 2 \frac{i}{2} - 3 (\frac{1}{2} + \frac{i}{2}) + 4$

Rewrite the expression.

$y = i - 3 (\frac{1}{2} + \frac{i}{2}) + 4$

Apply the distributive property.

$y = i - 3 (\frac{1}{2}) - 3 \frac{i}{2} + 4$

Combine $- 3$ and $\frac{1}{2}$ .

$y = i + \frac{- 3}{2} - 3 \frac{i}{2} + 4$

Combine $- 3$ and $\frac{i}{2}$ .

$y = i + \frac{- 3}{2} + \frac{- 3 i}{2} + 4$

Simplify each term.

Move the negative in front of the fraction.

$y = i - \frac{3}{2} + \frac{- 3 i}{2} + 4$

Move the negative in front of the fraction.

$y = i - \frac{3}{2} - \frac{3 i}{2} + 4$

To write $i$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = i \cdot \frac{2}{2} - \frac{3 i}{2} - \frac{3}{2} + 4$

Combine $i$ and $\frac{2}{2}$ .

$y = \frac{i \cdot 2}{2} - \frac{3 i}{2} - \frac{3}{2} + 4$

Simplify the expression.

Combine the numerators over the common denominator.

$y = \frac{- i}{2} - \frac{3}{2} + 4$

Move the negative in front of the fraction.

$y = - \frac{i}{2} - \frac{3}{2} + 4$

To write $4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = - \frac{i}{2} - \frac{3}{2} + 4 \cdot \frac{2}{2}$

Combine $4$ and $\frac{2}{2}$ .

$y = - \frac{i}{2} - \frac{3}{2} + \frac{4 \cdot 2}{2}$

Combine the numerators over the common denominator.

$y = - \frac{i}{2} + \frac{- 3 + 4 \cdot 2}{2}$

Simplify the numerator.

Multiply $4$ by $2$ .

$y = - \frac{i}{2} + \frac{- 3 + 8}{2}$

Add $- 3$ and $8$ .

$y = - \frac{i}{2} + \frac{5}{2}$

Reorder $- \frac{i}{2}$ and $\frac{5}{2}$ .

$y = \frac{5}{2} - \frac{i}{2}$

Step 4

Evaluate $y$ when $x = \frac{1}{2} - \frac{i}{2}$ .

Substitute $\frac{1}{2} - \frac{i}{2}$ for $x$ .

$y = 2 {(\frac{1}{2} - \frac{i}{2})}^{2} - 3 (\frac{1}{2} - \frac{i}{2}) + 4$

Simplify $2 {(\frac{1}{2} - \frac{i}{2})}^{2} - 3 (\frac{1}{2} - \frac{i}{2}) + 4$ .

Simplify each term.

Rewrite ${(\frac{1}{2} - \frac{i}{2})}^{2}$ as $(\frac{1}{2} - \frac{i}{2}) (\frac{1}{2} - \frac{i}{2})$ .

$y = 2 ((\frac{1}{2} - \frac{i}{2}) (\frac{1}{2} - \frac{i}{2})) - 3 (\frac{1}{2} - \frac{i}{2}) + 4$

Expand $(\frac{1}{2} - \frac{i}{2}) (\frac{1}{2} - \frac{i}{2})$ using the FOIL Method.

Apply the distributive property.

$y = 2 (\frac{1}{2} (\frac{1}{2} - \frac{i}{2}) - \frac{i}{2} (\frac{1}{2} - \frac{i}{2})) - 3 (\frac{1}{2} - \frac{i}{2}) + 4$

Apply the distributive property.

$y = 2 (\frac{1}{2} \cdot \frac{1}{2} + \frac{1}{2} (- \frac{i}{2}) - \frac{i}{2} (\frac{1}{2} - \frac{i}{2})) - 3 (\frac{1}{2} - \frac{i}{2}) + 4$

Apply the distributive property.

$y = 2 (\frac{1}{2} \cdot \frac{1}{2} + \frac{1}{2} (- \frac{i}{2}) - \frac{i}{2} \cdot \frac{1}{2} - \frac{i}{2} (- \frac{i}{2})) - 3 (\frac{1}{2} - \frac{i}{2}) + 4$

Simplify and combine like terms.

Simplify each term.

Multiply $\frac{1}{2} \cdot \frac{1}{2}$ .

Multiply $\frac{1}{2}$ by $\frac{1}{2}$ .

$y = 2 (\frac{1}{2 \cdot 2} + \frac{1}{2} (- \frac{i}{2}) - \frac{i}{2} \cdot \frac{1}{2} - \frac{i}{2} (- \frac{i}{2})) - 3 (\frac{1}{2} - \frac{i}{2}) + 4$

Multiply $2$ by $2$ .

$y = 2 (\frac{1}{4} + \frac{1}{2} (- \frac{i}{2}) - \frac{i}{2} \cdot \frac{1}{2} - \frac{i}{2} (- \frac{i}{2})) - 3 (\frac{1}{2} - \frac{i}{2}) + 4$

Multiply $\frac{1}{2} (- \frac{i}{2})$ .

Multiply $\frac{1}{2}$ by $\frac{i}{2}$ .

$y = 2 (\frac{1}{4} - \frac{i}{2 \cdot 2} - \frac{i}{2} \cdot \frac{1}{2} - \frac{i}{2} (- \frac{i}{2})) - 3 (\frac{1}{2} - \frac{i}{2}) + 4$

Multiply $2$ by $2$ .

$y = 2 (\frac{1}{4} - \frac{i}{4} - \frac{i}{2} \cdot \frac{1}{2} - \frac{i}{2} (- \frac{i}{2})) - 3 (\frac{1}{2} - \frac{i}{2}) + 4$

Multiply $- \frac{i}{2} \cdot \frac{1}{2}$ .

Multiply $\frac{1}{2}$ by $\frac{i}{2}$ .

$y = 2 (\frac{1}{4} - \frac{i}{4} - \frac{i}{2 \cdot 2} - \frac{i}{2} (- \frac{i}{2})) - 3 (\frac{1}{2} - \frac{i}{2}) + 4$

Multiply $2$ by $2$ .

$y = 2 (\frac{1}{4} - \frac{i}{4} - \frac{i}{4} - \frac{i}{2} (- \frac{i}{2})) - 3 (\frac{1}{2} - \frac{i}{2}) + 4$

Multiply $- \frac{i}{2} (- \frac{i}{2})$ .

Multiply $- 1$ by $- 1$ .

$y = 2 (\frac{1}{4} - \frac{i}{4} - \frac{i}{4} + 1 \frac{i}{2} \frac{i}{2}) - 3 (\frac{1}{2} - \frac{i}{2}) + 4$

Multiply $\frac{i}{2}$ by $1$ .

$y = 2 (\frac{1}{4} - \frac{i}{4} - \frac{i}{4} + \frac{i}{2} \cdot \frac{i}{2}) - 3 (\frac{1}{2} - \frac{i}{2}) + 4$

Multiply $\frac{i}{2}$ by $\frac{i}{2}$ .

$y = 2 (\frac{1}{4} - \frac{i}{4} - \frac{i}{4} + \frac{i i}{2 \cdot 2}) - 3 (\frac{1}{2} - \frac{i}{2}) + 4$

Raise $i$ to the power of $1$ .

$y = 2 (\frac{1}{4} - \frac{i}{4} - \frac{i}{4} + \frac{i^{1} i}{2 \cdot 2}) - 3 (\frac{1}{2} - \frac{i}{2}) + 4$

Raise $i$ to the power of $1$ .

$y = 2 (\frac{1}{4} - \frac{i}{4} - \frac{i}{4} + \frac{i^{1} i^{1}}{2 \cdot 2}) - 3 (\frac{1}{2} - \frac{i}{2}) + 4$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = 2 (\frac{1}{4} - \frac{i}{4} - \frac{i}{4} + \frac{i^{1 + 1}}{2 \cdot 2}) - 3 (\frac{1}{2} - \frac{i}{2}) + 4$

Add $1$ and $1$ .

$y = 2 (\frac{1}{4} - \frac{i}{4} - \frac{i}{4} + \frac{i^{2}}{2 \cdot 2}) - 3 (\frac{1}{2} - \frac{i}{2}) + 4$

Multiply $2$ by $2$ .

$y = 2 (\frac{1}{4} - \frac{i}{4} - \frac{i}{4} + \frac{i^{2}}{4}) - 3 (\frac{1}{2} - \frac{i}{2}) + 4$

Rewrite $i^{2}$ as $- 1$ .

$y = 2 (\frac{1}{4} - \frac{i}{4} - \frac{i}{4} + \frac{- 1}{4}) - 3 (\frac{1}{2} - \frac{i}{2}) + 4$

Move the negative in front of the fraction.

$y = 2 (\frac{1}{4} - \frac{i}{4} - \frac{i}{4} - \frac{1}{4}) - 3 (\frac{1}{2} - \frac{i}{2}) + 4$

Combine the numerators over the common denominator.

$y = 2 (\frac{1 - 1}{4} - \frac{i}{4} - \frac{i}{4}) - 3 (\frac{1}{2} - \frac{i}{2}) + 4$

Subtract $1$ from $1$ .

$y = 2 (\frac{0}{4} - \frac{i}{4} - \frac{i}{4}) - 3 (\frac{1}{2} - \frac{i}{2}) + 4$

Subtract $\frac{i}{4}$ from $- \frac{i}{4}$ .

$y = 2 (\frac{0}{4} - 2 \frac{i}{4}) - 3 (\frac{1}{2} - \frac{i}{2}) + 4$

Simplify each term.

Divide $0$ by $4$ .

$y = 2 (0 - 2 \frac{i}{4}) - 3 (\frac{1}{2} - \frac{i}{2}) + 4$

Cancel the common factor of $2$ .

Factor $2$ out of $- 2$ .

$y = 2 (0 + 2 (- 1) \frac{i}{4}) - 3 (\frac{1}{2} - \frac{i}{2}) + 4$

Factor $2$ out of $4$ .

$y = 2 (0 + 2 \cdot - 1 \frac{i}{2 \cdot 2}) - 3 (\frac{1}{2} - \frac{i}{2}) + 4$

Cancel the common factor.

$y = 2 (0 + 2 \cdot - 1 \frac{i}{2 \cdot 2}) - 3 (\frac{1}{2} - \frac{i}{2}) + 4$

Rewrite the expression.

$y = 2 (0 - 1 \frac{i}{2}) - 3 (\frac{1}{2} - \frac{i}{2}) + 4$

Rewrite $- 1 \frac{i}{2}$ as $- \frac{i}{2}$ .

$y = 2 (0 - \frac{i}{2}) - 3 (\frac{1}{2} - \frac{i}{2}) + 4$

Subtract $\frac{i}{2}$ from $0$ .

$y = 2 (- \frac{i}{2}) - 3 (\frac{1}{2} - \frac{i}{2}) + 4$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{i}{2}$ into the numerator.

$y = 2 \frac{- i}{2} - 3 (\frac{1}{2} - \frac{i}{2}) + 4$

Cancel the common factor.

$y = 2 \frac{- i}{2} - 3 (\frac{1}{2} - \frac{i}{2}) + 4$

Rewrite the expression.

$y = - i - 3 (\frac{1}{2} - \frac{i}{2}) + 4$

Apply the distributive property.

$y = - i - 3 (\frac{1}{2}) - 3 (- \frac{i}{2}) + 4$

Combine $- 3$ and $\frac{1}{2}$ .

$y = - i + \frac{- 3}{2} - 3 (- \frac{i}{2}) + 4$

Multiply $- 3 (- \frac{i}{2})$ .

Multiply $- 1$ by $- 3$ .

$y = - i + \frac{- 3}{2} + 3 \frac{i}{2} + 4$

Combine $3$ and $\frac{i}{2}$ .

$y = - i + \frac{- 3}{2} + \frac{3 i}{2} + 4$

Move the negative in front of the fraction.

$y = - i - \frac{3}{2} + \frac{3 i}{2} + 4$

To write $- i$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = - i \cdot \frac{2}{2} + \frac{3 i}{2} - \frac{3}{2} + 4$

Combine $- i$ and $\frac{2}{2}$ .

$y = \frac{- i \cdot 2}{2} + \frac{3 i}{2} - \frac{3}{2} + 4$

Combine the numerators over the common denominator.

$y = \frac{i}{2} - \frac{3}{2} + 4$

To write $4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{i}{2} - \frac{3}{2} + 4 \cdot \frac{2}{2}$

Combine $4$ and $\frac{2}{2}$ .

$y = \frac{i}{2} - \frac{3}{2} + \frac{4 \cdot 2}{2}$

Combine the numerators over the common denominator.

$y = \frac{i}{2} + \frac{- 3 + 4 \cdot 2}{2}$

Simplify the numerator.

Multiply $4$ by $2$ .

$y = \frac{i}{2} + \frac{- 3 + 8}{2}$

Add $- 3$ and $8$ .

$y = \frac{i}{2} + \frac{5}{2}$

Reorder $\frac{i}{2}$ and $\frac{5}{2}$ .

$y = \frac{5}{2} + \frac{i}{2}$

Step 5

List all of the solutions.

$y = \frac{5}{2} - \frac{i}{2}, x = \frac{1}{2} + \frac{i}{2}$

$y = \frac{5}{2} + \frac{i}{2}, x = \frac{1}{2} - \frac{i}{2}$

Step 6

Solve the System of Equations y=-x+3 y=2x^2-3x+4

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