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Through (3,13), parallel to 3x+5y=29

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Hi there! I'll help you solve this problem!

We are asked to find the equation of a line, given that:

  • The line is parallel to 3x + 5y = 29
  • The line passes through the point (3,13)

step one

So first, let's convert 3x + 5y = 29 to slope-intercept form.

Slope-intercept form

  • Slope intercept is a way of writing the equation of a line. It's formatted as
    y=mx+b where the parameter
    m defines the slope and
    b defines the y-intercept.

Basically, we need to convert the equation
3x+5y=29, which is in standard form, to
y=mx+b, which is slope intercept form.

So first, we subtract 3x from both sides:


5y=29-3x

Next, divide both sides by 5


y=\cfrac{29-3x}{5}

Rewrite it this way:

y=\cfrac{29}{5}-\cfrac{3x}{5}

Our equation is now in slope intercept form. The next step is to figure out the values of m and b.

  • m = -3/5
  • b = 29/5

step two

But that's the slope of the "old" line - what about the slope of the new line? Well, that we'll determine soon. Remember that the new line is parallel to y = 29/5 - 3x/5. Well, guess what? Parallel lines actually have the same slope! So the new line has the same slope as the line

y = 29/5 - 3x/5, which is -3/5.

step three

Now that we know the slope and the point that the line intersects, we can get down to writing our equation in point-slope form.

Point-slope form is
\boldsymbol{y-y_1=m(x-x_1)}.

Substitute the values:


  • \boldsymbol{y-13=-\cfrac{3}{5}(x-3)}}

Now we need to simplify that, and convert that to slope-intercept form!

step four

Distribute -3/5:


\boldsymbol{y-13=-\cfrac{3}{5}x-\cfrac{3}{1}*-\cfrac{3}{5}}

Simplify:


\boldsymbol{y-13=-\cfrac{3}{5}x+\cfrac{3*3}{5}}


\boldsymbol{y-13=-\cfrac{3}{5}x+\cfrac{9}{5}}

The next step is to add 13 to both sides:


\boldsymbol{y=-\cfrac{3}{5}x-\cfrac{9}{5}+\cfrac{13}{1}}

Uh oh! These fractions have different denominators - we can't add them just now! So, we need to write a common denominator:


\boldsymbol{y=-\cfrac{3}{5}x+\cfrac{9}{5}+\cfrac{13\cdot5}{5}}


\boldsymbol{y=-\cfrac{3}{5}x+\cfrac{9}{5}+\cfrac{65}{5}}


\boldsymbol{y=-\cfrac{3}{5}x+\cfrac{65+9}{5}}


\boxed{\boxed{\boldsymbol{y=-\cfrac{3}{5}x+\cfrac{74}{5}}}}

Therefore, the equation of the line that passes through the point (3,13) and is parallel to 3x + 5y = 29 is y = -3/5x + 74/5.

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Best wishes!

User Ayan Biswas
by
8.1k points
5 votes

Answer:

3x + 5y = 74

Explanation:

To find the equation of a line that passes through the point (3,13) and is parallel to the line 3x + 5y = 29, we can use the fact that two parallel lines have the same slope. The given line has the equation 3x + 5y = 29. We need to find the slope of this line, and then we can use it to write the equation of the parallel line.

First, let's rearrange the equation 3x + 5y = 29 to solve for y:

3x + 5y = 29

Subtract 3x from both sides:

3x + 5y - 3x = 29 - 3x

5y = -3x + 29

Now, divide both sides by 5 to isolate y:


\sf (5y)/(5)=-(3)/(5)x+(29)/(5)


\sf y=-(3)/(5)x+(29)/(5)

While comparing with y = mx + c, we get


\sf m =-(3)/(5)

So, the slope of the given line is
\sf -(3)/(5).

Now that we know the slope, we can use the point-slope form of a line to write the equation of the parallel line:


\sf y - y_1 = m(x - x_1)

Where (x1, y1) is the point (3,13) and m is the slope, which is -3/5.

Now, substitute the given value:


\sf y - 13 = -(3)/(5)(x - 3)

Now, distribute the slope and simplify:


\sf y - 13 = -(3)/(5)x + (9)/(5)

y - 13 = (-3/5)x + 9/5

To get rid of the fraction, we can multiply both sides of the equation by 5:


\sf 5(y - 13) = -3x + 9

Now, distribute the 5 on the left side:


\sf 5y - 65 = -3x + 9

Add 65 to both sides to isolate -3x:


\sf 5y - 65+65 = -3x + 9 + 65


5y = -3x + 74

Now, we can rewrite the equation in the standard form (Ax + By = C):

3x + 5y = 74

So, the equation of the line that passes through (3,13) and is parallel to 3x + 5y = 29 is 3x + 5y = 74.

User Savino Sguera
by
7.6k points

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