Solve the System of Equations y=-1/3x+7 y=-2x^3+5x^2+x-2

$y = - \frac{1}{3} x + 7$ $y = - 2 x^{3} + 5 x^{2} + x - 2$

Step 1

Eliminate the equal sides of each equation and combine.

$- \frac{1}{3} x + 7 = - 2 x^{3} + 5 x^{2} + x - 2$

Step 2

Solve $- \frac{1}{3} x + 7 = - 2 x^{3} + 5 x^{2} + x - 2$ for $x$ .

Combine $x$ and $\frac{1}{3}$ .

$- \frac{x}{3} + 7 = - 2 x^{3} + 5 x^{2} + x - 2$

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x \approx - 1.19709411 + y = - 2 x^{3} + 5 x^{2} + x - 2$

Step 3

Evaluate $y$ when $x = - 1.19709411$ .

Substitute $- 1.19709411$ for $x$ .

$y = - 2 {(- 1.19709411)}^{3} + 5 {(- 1.19709411)}^{2} - 1.19709411 - 2$

Substitute $- 1.19709411$ for $x$ in $y = - 2 {(- 1.19709411)}^{3} + 5 {(- 1.19709411)}^{2} - 1.19709411 - 2$ and solve for $y$ .

Remove parentheses.

$y = - 2 {(- 1.19709411)}^{3} + 5 {(- 1.19709411)}^{2} - 1.19709411 - 2$

Simplify $- 2 {(- 1.19709411)}^{3} + 5 {(- 1.19709411)}^{2} - 1.19709411 - 2$ .

Simplify each term.

Raise $- 1.19709411$ to the power of $3$ .

$y = - 2 \cdot - 1.71547694 + 5 {(- 1.19709411)}^{2} - 1.19709411 - 2$

Multiply $- 2$ by $- 1.71547694$ .

$y = 3.43095389 + 5 {(- 1.19709411)}^{2} - 1.19709411 - 2$

Raise $- 1.19709411$ to the power of $2$ .

$y = 3.43095389 + 5 \cdot 1.43303431 - 1.19709411 - 2$

Multiply $5$ by $1.43303431$ .

$y = 3.43095389 + 7.16517159 - 1.19709411 - 2$

Simplify by adding and subtracting.

Add $3.43095389$ and $7.16517159$ .

$y = 10.59612548 - 1.19709411 - 2$

Subtract $1.19709411$ from $10.59612548$ .

$y = 9.39903137 - 2$

Subtract $2$ from $9.39903137$ .

$y = 7.39903137$

Step 4

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(- 1.19709411, 7.39903137)$

Step 5

The result can be shown in multiple forms.

Point Form:

$(- 1.19709411, 7.39903137)$

Equation Form:

$x = - 1.19709411, y = 7.39903137$

Step 6