Parallel lines have the same slope. Solve for the slope in the first line by converting the equation to slope-intercept form.

3x + 4y = 12

4y = –3x + 12

y = –(3/4)x + 3

slope = –3/4

We know that the second line will also have a slope of –3/4, and we are given the point (1,2). We can set up an equation in slope-intercept form and use these values to solve for the y-intercept.

y = mx + b

2 = –3/4(1) + b

2 = –3/4 + b

b = 2 + 3/4 = 2.75

Plug the y-intercept back into the equation to get our final answer.

y = –(3/4)x + 2.75

To solve, we will need to find the slope of the line. We know that it is parallel to the line given by the equation, meaning that the two lines will have equal slopes. Find the slope of the given line by converting the equation to slope-intercept form.

\(\displaystyle 3x + 5y = 8\)

\(\displaystyle 5y=-3x+8\)

\(\displaystyle y=-\frac{3}{5}x+\frac{8}{5}\)

The slope of the line will be \(\displaystyle -\frac{3}{5}\). In slope intercept-form, we know that the line will be \(\displaystyle y=-\frac{3}{5}x+b\). Now we can use the given point to find the y-intercept.

\(\displaystyle y=-\frac{3}{5}x+b\)

\(\displaystyle 4=-\frac{3}{5}(10)+b\)

\(\displaystyle 4=-6+b\)

\(\displaystyle 10=b\)

The final equation for the line will be \(\displaystyle y=-\frac{3}{5}x+10\).

Start by converting the original equation to slop-intercept form.

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

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

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

The slope of this line is \(\displaystyle -\frac{3}{2}\). A parallel line will have the same slope. Now that we know the slope of our new line, we can use slope-intercept form and the given point to solve for the y-intercept.

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

\(\displaystyle -1=-\frac{3}{2}(2)+b\)

\(\displaystyle -1=-3+b\)

\(\displaystyle b=2\)

Plug the y-intercept into the slope-intercept equation to get the final answer.

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

The line parallel to \(\displaystyle \small y=\frac{1}{2}x+3\) must have a slope of \(\displaystyle \frac{1}{2}\), giving us the equation \(\displaystyle \small y=\frac{1}{2}x+b\). To solve for b, we can substitute the values for y and x.

\(\displaystyle \small 2=(\frac{1}{2})(4)+b\) 

\(\displaystyle \small 2=2+b\)

\(\displaystyle \small b=0\)

Therefore, the equation of the line is \(\displaystyle \small y=\frac{1}{2}x\).

Converting the given line to slope-intercept form we get the following equation:

\(\displaystyle y = \frac{-5}{3}x + \frac{8}{3}\)

For parallel lines, the slopes must be equal, so the slope of the new line must also be \(\displaystyle \frac{-5}{3}\). We can plug the new slope and the given point into the slope-intercept form to solve for the y-intercept of the new line.

\(\displaystyle y = mx + b\)

\(\displaystyle -4 = \frac{-5}{3}(6) + b\)

\(\displaystyle -4=-10+b\)

\(\displaystyle b = 6\)

Use the y-intercept in the slope-intercept equation to find the final answer.

\(\displaystyle y = \frac{-5}{3}x + 6\)

Find the slope of the given line: \(\displaystyle y=mx+b\) (slope intercept form)

\(\displaystyle y=\frac{-2}{3}x+2\) therefore the slope is \(\displaystyle \frac{-2}{3}\)

Parallel lines have the same slope, so now we need to find the equation of a line with slope \(\displaystyle \frac{-2}{3}\) and going through point \(\displaystyle (3, 2)\) by substituting values into the point-slope formula.

\(\displaystyle 2=\frac{-2}{3}(3)+b\)

So, \(\displaystyle b=4\)

Thus, the new equation is \(\displaystyle y=\frac{-2}{3}x+4\)

This is a two-step problem. First, the slope of the original line needs to be found. The slope will be represented by "\(\displaystyle m\)" when the line is in \(\displaystyle y\)-intercept form \(\displaystyle (y=mx+b)\).

\(\displaystyle 2x +3y = 16\)

\(\displaystyle 3y = -2x +16\)

\(\displaystyle y = \left(\frac{-2}{3}\right)x +\left(\frac{16}{3}\right)\)

So the slope of the original line is \(\displaystyle -\frac{2}{3}\). A line with perpendicular slope will have a slope that is the inverse reciprocal of the original. So in this case, the slope would be \(\displaystyle \frac{3}{2}\). The second step is finding which line will give you that slope. For the correct answer, we find the following:

\(\displaystyle 6y - 9x = 16.5\)

\(\displaystyle 6y = 9x +16.5\)

\(\displaystyle y= \left(\frac{3}{2}x\right) +\left(\frac{16.5}{6}\right)\)

So, the slope is \(\displaystyle \frac{3}{2}\), and this line is perpendicular to the original.

Since parallel lines have equal slopes, you should find the slope of the line given to you. The easiest way to do this is to solve the equation so that its form is \(\displaystyle y=mx+b\).  \(\displaystyle m\) represents the slope.

Take your equation: \(\displaystyle 4x + 7y = 49 - 2x + 4y\)

First, subract \(\displaystyle 4y\) from both sides:

\(\displaystyle 4x + 3y = 49 - 2x\)

Next, subtract \(\displaystyle 4x\) from both sides:

\(\displaystyle 3y = 49 - 6x\)

Finally, divide by \(\displaystyle 3\):

\(\displaystyle y = \frac{49}{3} - 2x\), which is the same as \(\displaystyle y = - 2x + \frac{49}{3}\)

Thus, your slope is \(\displaystyle -2\).

Among the options provided only \(\displaystyle 2y + x = 40 - 3x\) is parallel. Solve this equation as well for \(\displaystyle y=mx+b\) form.  

First, subtract \(\displaystyle x\) from both sides:

\(\displaystyle 2y = 40 - 4x\)

Then, divide by \(\displaystyle 2\):

\(\displaystyle y = 20 - 2x\)

Parallel lines have the same slope but different y-intercepts. When the equations of two lines are the same they have infinitely many points in common, whereas parallel lines have no points in common.

Our equation is given in slope-intercept form,

\(\displaystyle y=mx+b\)

where \(\displaystyle m\) is the slope. In this particular situation \(\displaystyle m=-2\).

Therefore we want to find an equation that has the same \(\displaystyle m\) value and a different \(\displaystyle b\) value.

Thus,

\(\displaystyle y = -2x -1\) is parallel to our equation.

Parallel lines have the same slope and differing y-intercepts. Since \(\displaystyle y = 9x + 5\) is the only equation with the same slope, and the y-intercept is different, this is the equation of the parallel line.