Rewrite the given equation in standard form to slope-intercept form.

$\displaystyle y=mx+b$

Subtract $\displaystyle 2x$ from both sides.

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

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

Divide by three on both sides.

$\displaystyle \frac{3y}{3}=\frac{-2x+6}{3}$

Simplify this equation.

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

Shifting this equation down means that the y-intercept will be subtracted six.

$\displaystyle y=-\frac{2}{3}x+2-6$

The answer is:  $\displaystyle y=-\frac{2}{3}x-4$

The given equation can be rewritten in slope-intercept format, $\displaystyle y=mx+b$.

$\displaystyle y=10-2x \rightarrow y=-2x+10$

Shifting down a line fifteen units will decrease the y-intercept by 15.

$\displaystyle y=-2x+10-15$

The answer is:  $\displaystyle y=-2x-5$

Shifting the equation up one unit will change the y-intercept by adding one.

$\displaystyle y=2x+5+(1) = 2x+6$

If the graph is to be shifted four units to the left, the x-variable will need to be replaced with the quantity $\displaystyle (x+4)$.

$\displaystyle y=2(x+4)+6$

Use the distribution property to simplify the binomial.

$\displaystyle y=2x+8+6 = 2x+14$

The equation is:  $\displaystyle y= 2x+14$

Rewrite this equation in standard form to slope intercept format.

Subtract $\displaystyle x$ on both sides.

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

$\displaystyle 3y=-x+6$

Divide by three on both sides.

$\displaystyle \frac{3y}{3}=\frac{-x+6}{3}$

Simplify this equation.

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

If this graph is shifted up four units, simply add four to the y-intercept.

$\displaystyle y=-\frac{1}{3}x+2+4$

The answer is: $\displaystyle y=-\frac{1}{3}x+6$

Shifting the graph left 8 units will require changing the x-variable to $\displaystyle (x+8)$.

Replace the term and simplify the equation.

$\displaystyle y=6(x+8)-4$

Distribute the six through the binomial.

$\displaystyle y=6x+48-4 = 6x+44$

The equation is:  $\displaystyle y= 6x+44$

If the graph is shifted leftward, apply the transformation by replacing the x-variable with $\displaystyle (x+5)$.

$\displaystyle y=3-9(x+5)$

Simplify this equation by distribution, and rewrite this in slope intercept format.

$\displaystyle y=3-9x-45$

Combine like-terms.

The answer is:  $\displaystyle y=-9x-42$

Rewrite the equation in slope-intercept form:  $\displaystyle y=mx+b$

$\displaystyle y=-3x+7$

Shifting this line down three units will decrease the y-intercept by three.

$\displaystyle y=-3x+7-(3) = -3x+4$

$\displaystyle y= -3x+4$

If the line is shifted left four units, the x-variable will need to be replaced with $\displaystyle (x+4)$.

$\displaystyle y= -3(x+4)+4$

Simplify this equation.

$\displaystyle y=-3x-12+4 = -3x-8$

The answer is:  $\displaystyle y= -3x-8$

Rewrite the given equation in standard form to slope-intercept form.

Add $\displaystyle y$ on both sides.

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

$\displaystyle y=x$

A shift down four units will decrease the y-intercept by four.  The current y-intercept is zero.  Rewrite the equation.

$\displaystyle y=x-4$

The line shifted three units to the left means that the x-variable will need to be replaced with $\displaystyle (x+3)$.

Rewrite the equation.

$\displaystyle y=(x+3)-4$

Simplify this equation.

The answer is:  $\displaystyle y=x-1$

Simplify the equation by distribution.

$\displaystyle y=-3(-x+8) = (-3)(-x)+(-3)(8) = 3x-24$

Shifting this line up by three units will add three to the y-intercept.

$\displaystyle y= 3x-24+3 = 3x-21$

The answer is:  $\displaystyle y= 3x-21$

Use distribution to simplify this equation.  We will need to put the equation in slope-intercept format, $\displaystyle y=mx+b$

$\displaystyle y=2(2x+3) =2(2x)+2(3) =4x+6$

The equation becomes:  $\displaystyle y =4x+6$

Shifting this equation up five units will add five to the y-intercept.

The answer is:  $\displaystyle y =4x+11$