The $\displaystyle y$-intercept of the line can be seen to be at the point five units above the origin, which is $\displaystyle (0, 5)$. The $\displaystyle x$-intercept is at the point three units to the right of the origin, which is $\displaystyle (3,0)$. From these intercepts, we can find slope $\displaystyle m$ by setting $\displaystyle b= 5,a = 3$ in the formula

$\displaystyle m = - \frac{b}{a}$

The slope is

$\displaystyle m = - \frac{5}{3}$

Now, we can find the slope-intercept form of the line 

$\displaystyle y = mx + b$

By setting $\displaystyle m = - \frac{5}{3}$, $\displaystyle b= 5$:

$\displaystyle y = - \frac{5}{3}x +5$

The standard form of a linear equation in two variables is

$\displaystyle ax+by = c$,

so in order to find the equation in this form, first, add $\displaystyle - \frac{5}{3}x$ to both sides:

$\displaystyle y + \frac{5}{3}x = - \frac{5}{3}x +5 + \frac{5}{3}x$

$\displaystyle \frac{5}{3}x+ y = 5$

We can eliminate the fraction by multiplying both sides by 3:

$\displaystyle 3 \left (\frac{5}{3}x+ y \right )= 3 \cdot 5$

Distribute by multiplying:

$\displaystyle 3 \cdot \frac{5}{3}x+3 \cdot y =15$

$\displaystyle 5x+3y = 15$,

the correct equation.

The equation in standard form is:  $\displaystyle Ax+By =C$

Simplify the right side by distribution.

$\displaystyle 3+y =2(2x)-(2)(6)$

$\displaystyle y+3 = 4x-12$

Subtract $\displaystyle 4x$ on both sides.

$\displaystyle y+3 -4x= 4x-12-4x$

The equation becomes: 

$\displaystyle -4x+y+3 = -12$

Subtract 3 from both sides.

$\displaystyle -4x+y+3-3 = -12-3$

The answer is:  $\displaystyle -4x+y = -15$

We will first need to write the point-slope form to set up the equation.

$\displaystyle y-y_1 = m(x-x_1)$

Substitute the slope and point.

$\displaystyle y-3= 2(x-1)$

Simplify the right side.

$\displaystyle y-3 = 2x-2$

Add 3 on both sides.

$\displaystyle y-3+3 = 2x-2+3$

$\displaystyle y=2x+1$

Subtract $\displaystyle 2x$ on both sides.

$\displaystyle y-2x=2x+1-2x$

The answer is:  $\displaystyle -2x+y = 1$

Distribute the right side.

$\displaystyle y=4x-16$

Subtract $\displaystyle 4x$ on both sides.

$\displaystyle y-4x=4x-16-4x$

The answer is:  $\displaystyle -4x+y = -16$

The standard form of a linear equation is:  $\displaystyle Ax+By=C$

Multiply by two on both sides to eliminate the fraction.

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

$\displaystyle x= 4x-4y+6$

Subtract $\displaystyle x$ on both sides.

$\displaystyle x-x= 4x-4y+6-x$

$\displaystyle 0 = 4x-4y +6$

Subtract 6 from both sides.

The answer is:  $\displaystyle 4x-4y =-6$

The standard form of a line is:  $\displaystyle Ax+By =C$

First, we can write the equation in slope-intercept form:  $\displaystyle y=mx+b$

$\displaystyle y=3x+6$

Subtract $\displaystyle 3x$ on both sides.

$\displaystyle y-3x=3x+6-3x$

The answer is:  $\displaystyle -3x+y= 6$

The standard form of a line is:  $\displaystyle Ax+By =C$

We can use the point-slope form of a line since we are only given the slope and a point.

$\displaystyle y-y_1 = m( x-x_1)$

Substitute the slope and the point.

$\displaystyle y-5 = 7( x-2)$

Simplify this equation.

$\displaystyle y-5 = 7x-14$

Add $\displaystyle 5$ on both sides.

$\displaystyle y-5+5= 7x-14+5$

$\displaystyle y=7x-9$

Subtract $\displaystyle 7x$ from both sides.

$\displaystyle y- 7x=7x-9- 7x$

Simplify both sides.

$\displaystyle -7x+y =-9$

The answer is:  $\displaystyle -7x+y =-9$

Distribute the four through both terms of the binomial.

$\displaystyle y=4x-8$

Subtract $\displaystyle 4x$ on both sides of the equation.

$\displaystyle y-4x=4x-8-4x$

Simplify both sides.

$\displaystyle -4x+y = -8$

The answer is:  $\displaystyle -4x+y = -8$

First, find the slope-intercept form of the equation. This is 

$\displaystyle y = mx+b$,

where $\displaystyle m$ is the slope and $\displaystyle (0,b)$ is the $\displaystyle y$-intercept of the line. Since $\displaystyle (0, 8 )$ is this intercept, $\displaystyle b = 8$. Also, the slope of a line with intercepts $\displaystyle (a,0 )$ and $\displaystyle (0,b)$ is $\displaystyle m = - \frac{b}{a}$, so, setting $\displaystyle a = -3, b = 8$,

$\displaystyle m = - \frac{8}{-3} = \frac{8}{3}$.

The slope-intercept form is 

$\displaystyle y = \frac{8}{3} x + 8$

The standard form of the equation is 

$\displaystyle Ax+By = C$,

where, by custom, $\displaystyle A$, $\displaystyle B$, and $\displaystyle C$ are relatively prime integers, and $\displaystyle A > 0$. To accomplish this:

Switch the expressions:

$\displaystyle \frac{8}{3} x + 8 = y$

Add $\displaystyle -y-8$ to both sides:

$\displaystyle \frac{8}{3} x + 8+ (-y-8 ) = y + (-y-8 )$

$\displaystyle \frac{8}{3} x - y =-8$

Multiply both sides by 3 to eliminate the denominator and make the coefficients integers with GCF 1:

$\displaystyle 3 \cdot \left (\frac{8}{3} x - y \right ) =3 \cdot( -8 )$

Distribute on the left:

$\displaystyle 3 \cdot \frac{8}{3} x - 3 \cdot y =3 \cdot( -8 )$

$\displaystyle 8 x - 3 y =-24$

This is the correct equation.

The standard form of the equation of a line is

$\displaystyle ax+ by = c$.

To rewrite the equation

$\displaystyle y = \frac{2}{7}x- \frac{1}{14}$

in this form so that $\displaystyle x$ has a positive coefficient, first, switch the places of the expressions:

$\displaystyle \frac{2}{7}x- \frac{1}{14} = y$

Get the $\displaystyle y$ term on the left and the constant on the right by adding $\displaystyle -y+ \frac{1}{14}$ to both sides:

$\displaystyle \frac{2}{7}x- \frac{1}{14} + \left ( -y+ \frac{1}{14} \right ) = y+ \left ( -y+ \frac{1}{14} \right )$

$\displaystyle \frac{2}{7}x- y = \frac{1}{14}$

To eliminate fractions and ensure that the coefficients are  relatively prime, multiply both sides by lowest common denominator 14:

$\displaystyle 14 \cdot \left (\frac{2}{7}x- y \right ) = 14 \cdot \frac{1}{14}$

$\displaystyle 14 \left (\frac{2}{7}x- y \right ) = \frac{14}{1} \cdot \frac{1}{14}$

$\displaystyle 14 \cdot \left (\frac{2}{7}x- y \right ) =1$

Multiply 14 by both expressions in the parentheses:

$\displaystyle 14 \cdot \frac{2}{7}x-14 \cdot y =1$

$\displaystyle \frac{14}{1} \cdot \frac{2}{7}x-14y =1$

Cross-canceling:

$\displaystyle \frac{2}{1} \cdot \frac{2}{1}x-14y =1$

$\displaystyle 4x-14y =1$,

the correct choice.