The given line can be rewritten as \(\displaystyle y=-\frac{1}{2}x+\frac{7}{10}\), which has slope \(\displaystyle m=-\frac{1}{2}\).

If the new line is parallel to the old line, it must have the same slope.  So we use the point-slope form of an equation to calculate the new intercept.

\(\displaystyle y=mx+b\) becomes \(\displaystyle 7=-\frac{1}{2}(-6)+b\) where \(\displaystyle b=4\).

So the equation of the parallel line is \(\displaystyle y=-\frac{1}{2}x+4\).

Since parallel lines share the same slope, the only answer that works is \(\displaystyle y=12x+2.\)

In order for two lines to be parallel, they must have the same slope. The slope of the given line is \(\displaystyle 2\), so we know that the line going through the given point also has to have a slope of \(\displaystyle 2\). Using the point-slope formula,

\(\displaystyle y-y_{1}=m(x-x_1)\),

where \(\displaystyle m\) represents the slope and \(\displaystyle y_1\) and \(\displaystyle x_1\) represent the given points, plug in the points given and simplify into standard form:

\(\displaystyle y-4=2(x-(-2))\)

\(\displaystyle y-4=2x+4\)

\(\displaystyle y=2x+8\)

Parallel lines have the same slopes.  The slope for the given equation is \(\displaystyle 2\).  We can use the slope and the new point in the slope intercept equation to solve for the intercept:

\(\displaystyle y=mx+b\) 

\(\displaystyle -9=2(3)+b\) 

\(\displaystyle b=-15\)

Therefore the new equation becomes:

\(\displaystyle y=2x-15\)

Parallel lines have the same slope.  The slope of the given line is \(\displaystyle \frac{1}{2}\).

Find the line with slope \(\displaystyle \frac{1}{2}\) through the point \(\displaystyle (6,7)\) by plugging this informatuon into the slope intercept equation, \(\displaystyle y=mx+b\):

\(\displaystyle 7=\frac{1}{2}(6)+b\), which gives \(\displaystyle 7=3+b\).

Solve for \(\displaystyle b\) by subtracting \(\displaystyle 3\) from both sides to get \(\displaystyle b=4\).

Then the parallel line equation becomes \(\displaystyle y=\frac{1}{2}x+4\), and converting to standard form gives \(\displaystyle -x+2y=8\).

Parallel lines share the same slope. Because the slope of the original line is \(\displaystyle -\frac{3}{2}\), the correct answer must have that slope, so the correct answer is

\(\displaystyle y=-\frac{3}{2}X+7\)

The coordinates form a triangle in the second quadrant with a side along the y-axis.  The rotation about the origin by \(\displaystyle 90\) degrees clockwise results in a triangle in the first quadrant with a side along the x-axis.  There are two responses that give triangles along the x-axis: 

\(\displaystyle (3,0), \; (5,0),\; and \; (4,2)\) and

\(\displaystyle (6,0), \; (10,0),\; and \; (8,4)\)

A rotation and a dialation by a factor of \(\displaystyle 2\) is given by

\(\displaystyle (6,0), \; (10,0),\; and \; (8,4)\), so the correct answer is \(\displaystyle (3,0), \; (5,0),\; and \; (4,2)\)

To find the y-intercept, we set the \(\displaystyle x\) value equal to zero and solve for the value of \(\displaystyle y\).

\(\displaystyle y=5x+5\)

\(\displaystyle y=5(0)+5\)

\(\displaystyle y=0+5\)

\(\displaystyle y=5\)

Since the y-intercept is a point, we want to write our answer in point notation: \(\displaystyle (0,5)\).

To find the x-intercept of an equation, set the \(\displaystyle y\) value equal to zero and solve for \(\displaystyle x\).

\(\displaystyle y=\frac{2}{3}x+ 12\)

\(\displaystyle 0 =\frac{2}{3}x+12\)

Subtract \(\displaystyle 12\) from both sides.

\(\displaystyle 0 -12 = \frac{2}{3}x +12 - 12\)

\(\displaystyle -12 = \frac{2}{3} x\)

Multiply both sides by \(\displaystyle \frac{3}{2}\) .

\(\displaystyle (\frac{3}{2})(-12) = (\frac{2}{3} x)(\frac{3}{2})\)

\(\displaystyle -18= x\)

Since the x-intercept is a point, we will want to write it in point notation: \(\displaystyle (-18,0\)\(\displaystyle )\)

To solve for the y-intercept, set the x value equal to zero:

\(\displaystyle y=5x+20\)

\(\displaystyle y=5(0)+20\)

\(\displaystyle y=20\)