Start by rewriting the equation into slope-intercept form.

\(\displaystyle 17y+34x=2\)

\(\displaystyle 17y=-34x+2\)

\(\displaystyle y=-2x+\frac{2}{17}\)

To find which point is on the line, take the \(\displaystyle x\)-coordinate, and plug it into the given equation to solve for \(\displaystyle y\). If the \(\displaystyle y\)-value matches the \(\displaystyle y\)-coordinate of the same point, then the point is on the line.

Plugging in \(\displaystyle x=6\) into the given equation will give the following:

\(\displaystyle y=-2x+\frac{2}{17}\)

\(\displaystyle y=-2(6)+\frac{2}{17}=-\frac{202}{17}\)

Thus, \(\displaystyle (6, -\frac{202}{17})\) is on the line.

Start by rewriting the equation into slope-intercept form.

\(\displaystyle 6x+y=10\)

\(\displaystyle y=-6x+10\)

To find which point is on the line, take the \(\displaystyle x\)-coordinate, and plug it into the given equation to solve for \(\displaystyle y\). If the \(\displaystyle y\)-value matches the \(\displaystyle y\)-coordinate of the same point, then the point is on the line.

Plugging in \(\displaystyle x=-\frac{1}{2}\) into the given equation will give the following:

\(\displaystyle y=-6x+10\)

\(\displaystyle y=-6(-\frac{1}{2})+10=3+10=13\)

Thus, \(\displaystyle (-\frac{1}{2}, 13)\) is on the line.

Start by rewriting the equation into slope-intercept form.

\(\displaystyle -5x-2y=16\)

\(\displaystyle -2y=5x+16\)

\(\displaystyle y=-\frac{5}{2}x-8\)

To find which point is on the line, take the \(\displaystyle x\)-coordinate, and plug it into the given equation to solve for \(\displaystyle y\). If the \(\displaystyle y\)-value matches the \(\displaystyle y\)-coordinate of the same point, then the point is on the line.

Plugging in \(\displaystyle x=-6\) into the given equation will give the following:

\(\displaystyle y=-\frac{5}{2}x-8\)

\(\displaystyle y=-\frac{5}{2}(-6)-8=15-8=7\)

Thus, \(\displaystyle (-6, 7)\) is on the line.

A line of an equation passes through the point with coordinates \(\displaystyle (100,100)\) if and only if, when \(\displaystyle x=100, y=100\), the equation is true. Substitute for \(\displaystyle x\) and \(\displaystyle y\):

\(\displaystyle x+y = 100\)

\(\displaystyle 100+100 = 100\)

\(\displaystyle 200 = 100\) - this is false.

The line does not pass through the point.

The coordinates of the origin are \(\displaystyle (0,0)\), so the line of an equation passes through this point of and only if \(\displaystyle (0,0)\) is a solution of the equation - or, equivalently, if and only if setting \(\displaystyle x = 0\) and \(\displaystyle y= 0\) makes the equation a true statement. Substitute both values:

\(\displaystyle 3x= 4y\)

\(\displaystyle 3\cdot 0 = 4 \cdot 0\)

\(\displaystyle 0 = 0\)

The statement is true, so the line does pass through the origin.

If two distinct lines intersect at the point \(\displaystyle (7, -2)\) - that is, if both pass through this point - it follows that \(\displaystyle (7, -2)\) is a solution of the equations of both. Therefore, set \(\displaystyle x = 7, y = -2\) in the equations and determine whether they are true or not.

Examine the second equation:

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

\(\displaystyle 2 (7)+ 5(-2) = 6\)

\(\displaystyle 14+ (-10) = 6\)

\(\displaystyle 4 = 6\)

False; \(\displaystyle (7, -2)\) is not on the line of this equation.

Therefore, the lines cannot intersect at \(\displaystyle (7, -2)\).

If we have two points, we can find the slope of the line between them by using the definition of the slope:

\(\displaystyle slope = \frac{\Delta Y}{\Delta X}\)    where the triangle is the greek letter 'Delta', and is used as a symbol for 'difference' or 'change in'

Now that we have our slope ( \(\displaystyle \frac{4}{2}\), simplified to \(\displaystyle 2\)), we can write the equation for slope-intercept form:

\(\displaystyle y=mx+b\)   where \(\displaystyle m\) is the slope and \(\displaystyle b\) is the y-intercept

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

In order to find the y-intercept, we simply plug in one of the points on our line

\(\displaystyle (1)=2(-1)+b\)

\(\displaystyle 1=-2+b\)

\(\displaystyle b=3\)

So our equation looks like    \(\displaystyle y=2x+3\)

Because we have the desired slope and the y-intercept, we can easily write this as an equation in slope-intercept form (y=mx+b).

This gives us \(\displaystyle y=4x/3-5\). Because this does not match either of the answers in this form (y=mx+b), we must solve the equation for x. Adding 5 to each side gives us \(\displaystyle y+5=4x/3\). We can then multiple both sides by 3 and divide both sides by 4, giving us \(\displaystyle 3y/4+15/4=x\).

If the y-intercept of a line is \(\displaystyle 5\), then the \(\displaystyle y\)-value is \(\displaystyle 5\) when \(\displaystyle x\) is zero. Write the point:

\(\displaystyle (0,5)\)

If the \(\displaystyle x\)-intercept of a line is \(\displaystyle -1\), then the \(\displaystyle x\)-value is \(\displaystyle -1\) when \(\displaystyle y\) is zero. Write the point:

\(\displaystyle (-1,0)\)

Use the following formula for slope and the two points to determine the slope:

\(\displaystyle m=\frac{y_2-y_1}{x_2-x_1} = \frac{0-5}{-1-0}= \frac{-5}{-1} =5\)

Use the slope intercept form and one of the points, suppose \(\displaystyle (0,5)\), to find the equation of the line by substituting in the values of the point and solving for \(\displaystyle b\), the \(\displaystyle y\)-intercept.

\(\displaystyle y=mx+b\)

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

\(\displaystyle b=5\)

Therefore, the equation of this line is \(\displaystyle y=5x+5\).

The slope intercept form can be written as:

\(\displaystyle y=mx+b\)

where \(\displaystyle m\) is the slope and \(\displaystyle b\) is the y-intercept. Plug in the values of the slope and \(\displaystyle y\)-intercept into the equation.

The correct answer is: \(\displaystyle y=6x+6\)