Question

Parallel and Perpendicular Worksheet 2 ,2. Write the equation of the line that it is parallel to y = 2x + 1 and passes throug solution of the following system of equations. (3x - 2y - 10 x+y=5

Answer 1

To find the equation of a line parallel to y = 2x + 1 and passing through a solution of a system of equations, we can first determine that the parallel line will also have a slope of 2. By substituting the given solution into the equation of the parallel line, we can determine the equation of the line. In this case, the equation of the line is y = 2x.

Step-by-step explanation:

To find the equation of a line that is parallel to y = 2x + 1 and passes through a point, we need to use the fact that parallel lines have the same slope. The slope of the given line is 2, so the parallel line will also have a slope of 2. Let's call the equation of the parallel line y = 2x + b, where b is the y-intercept.

Since the line passes through a solution of the system of equations 3x - 2y - 10 = 0 and x + y = 5, substitute these values into the equation of the parallel line:

3x - 2y - 10 = 0 --> 3x - 2(5 - x) - 10 = 0

Simplify the equation and solve for x:

3x - 10 + 2x + 10 = 0 --> 5x = 0 --> x = 0

Substitute x = 0 into the equation of the parallel line to find the y-intercept:

y = 2(0) + b --> y = b

Therefore, the equation of the line that is parallel to y = 2x + 1 and passes through the solution of the system of equations is y = 2x.

Answer 2

y = 2x - 7

EXPLANATION

We have that the line is parallel to y = 2x + 1 and it passes through the solution of:

3x - 2y = 10

x + y = 5

Parallel lines have the same slope.

So, the slope of the line we are looking for will be the same as the slope of y = 2x + 1.

Linear equations are given generally as:

y = mx + c

where m = slope, c = y intercept

This means that the slope of the line is 2.

Now, we need to find the solution of the smultaneous equations given to know the point it passes through.

We have:

3x - 2y = 10

x + y = 5

From the second equation:

x = 5 - y

Put that in the first:

3(5 - y) - 2y = 10

15 - 3y - 2y = 10

Collect like terms:

-5y = 10 - 15 = -5

=> y = -5 / -5

y = 1

Therefore:

x = 5 - y = 5 - 1

x = 4

Therefore, the line passes through (4, 1)

We can now use the point-slope method to find the equation of the line:

`\begin{gathered} y-y_1=m(x-x_1) \\ \text{where m = 2 and (x1, y1) = (4, 1)} \\ \Rightarrow\text{ y - 1 = 2(x - 4) = 2x - 8} \\ \text{Collect like terms:} \\ y\text{ = 2x - 8 + 1} \\ y\text{ = 2x - 7} \end{gathered}`

That is the equation of the line.