Find the Equation Using Point-Slope Formula (1,-1) , y=3/2x+5

$(1, - 1)$ , $y = \frac{3}{2} x + 5$

Step 1

Use the slope $\frac{3}{2} x + 5$ and a given point $(1, - 1)$ to substitute for $x_{1}$ and $y_{1}$ in the point-slope form $y - y_{1} = m (x - x_{1})$ , which is derived from the slope equation $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 1) = (\frac{3}{2} x + 5) \cdot (x - (1))$

Step 2

Simplify the equation and keep it in point-slope form.

$y + 1 = (\frac{3}{2} x + 5) \cdot (x - 1)$

Step 3

Solve for $y$ .

Simplify $(\frac{3}{2} x + 5) \cdot (x - 1)$ .

Rewrite.

$y + 1 = 0 + 0 + (\frac{3}{2} x + 5) \cdot (x - 1)$

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

$y + 1 = (\frac{3 x}{2} + 5) \cdot (x - 1)$

Expand $(\frac{3 x}{2} + 5) (x - 1)$ using the FOIL Method.

Apply the distributive property.

$y + 1 = \frac{3 x}{2} (x - 1) + 5 (x - 1)$

Apply the distributive property.

$y + 1 = \frac{3 x}{2} x + \frac{3 x}{2} \cdot - 1 + 5 (x - 1)$

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $\frac{3 x}{2} x$ .

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

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

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

$y + 1 = \frac{3 (x^{1} x)}{2} + \frac{3 x}{2} \cdot - 1 + 5 x + 5 \cdot - 1$

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

$y + 1 = \frac{3 (x^{1} x^{1})}{2} + \frac{3 x}{2} \cdot - 1 + 5 x + 5 \cdot - 1$

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

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

Add $1$ and $1$ .

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

Multiply $\frac{3 x}{2} \cdot - 1$ .

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

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

Multiply $- 1$ by $3$ .

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

Move the negative in front of the fraction.

$y + 1 = \frac{3 x^{2}}{2} - \frac{(3) x}{2} + 5 x + 5 \cdot - 1$

Multiply $5$ by $- 1$ .

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

To write $5 x$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Combine $5 x$ and $\frac{2}{2}$ .

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

Combine the numerators over the common denominator.

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

Combine the numerators over the common denominator.

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

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Combine $- 5$ and $\frac{2}{2}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $2$ by $5$ .

$y + 1 = \frac{3 x^{2} - 3 x + 10 x - 5 \cdot 2}{2}$

Multiply $- 5$ by $2$ .

$y + 1 = \frac{3 x^{2} - 3 x + 10 x - 10}{2}$

Add $- 3 x$ and $10 x$ .

$y + 1 = \frac{3 x^{2} + 7 x - 10}{2}$

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot - 10 = - 30$ and whose sum is $b = 7$ .

Factor $7$ out of $7 x$ .

$y + 1 = \frac{3 x^{2} + 7 (x) - 10}{2}$

Rewrite $7$ as $- 3$ plus $10$

$y + 1 = \frac{3 x^{2} + (- 3 + 10) x - 10}{2}$

Apply the distributive property.

$y + 1 = \frac{3 x^{2} - 3 x + 10 x - 10}{2}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$y + 1 = \frac{(3 x^{2} - 3 x) + 10 x - 10}{2}$

Factor out the greatest common factor (GCF) from each group.

$y + 1 = \frac{3 x (x - 1) + 10 (x - 1)}{2}$

Factor the polynomial by factoring out the greatest common factor, $x - 1$ .

$y + 1 = \frac{(x - 1) (3 x + 10)}{2}$

Subtract $1$ from both sides of the equation.

$y = \frac{(x - 1) (3 x + 10)}{2} - 1$

Step 4

List the equation in different forms.

Slope-intercept form:

$y = \frac{(x - 1) (3 x + 10)}{2} - 1$

Point-slope form:

$y + 1 = (\frac{3}{2} x + 5) \cdot (x - 1)$

Step 5

Find the Equation Using Point-Slope Formula (1,-1) , y=3/2x+5

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