Solve left(4+yright)^2-left(1-yright)^2=1^2+5y

$${ \left(4+y \right) }^{ 2 } - { \left(1-y \right) }^{ 2 } = { 1 }^{ 2 } +5y$$

Solve for y

$$y = -\frac{14}{5} = -2\frac{4}{5} = -2.8$$

Steps for Solving Linear Equation

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(4+y\right)^{2}.

$$16+8y+y^{2}-\left(1-y\right)^{2}=1^{2}+5y$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(1-y\right)^{2}.

$$16+8y+y^{2}-\left(1-2y+y^{2}\right)=1^{2}+5y$$

To find the opposite of 1-2y+y^{2}, find the opposite of each term.

$$16+8y+y^{2}-1+2y-y^{2}=1^{2}+5y$$

Subtract 1 from 16 to get 15.

$$15+8y+y^{2}+2y-y^{2}=1^{2}+5y$$

Combine 8y and 2y to get 10y.

$$15+10y+y^{2}-y^{2}=1^{2}+5y$$

Combine y^{2} and -y^{2} to get 0.

$$15+10y=1^{2}+5y$$

Calculate 1 to the power of 2 and get 1.

$$15+10y=1+5y$$

Subtract 5y from both sides.

$$15+10y-5y=1$$

Combine 10y and -5y to get 5y.

$$15+5y=1$$

Subtract 15 from both sides.

$$5y=1-15$$

Subtract 15 from 1 to get -14.

$$5y=-14$$

Divide both sides by 5.

$$y=\frac{-14}{5}$$

Fraction \frac{-14}{5} can be rewritten as -\frac{14}{5} by extracting the negative sign.

$$y=-\frac{14}{5}$$