Find the equation of the line that passes through (1,3) and is perpendicular to y=2x+3. Leave your answer in the form ax+by=c Where a, b and c are integers.

To find the slope of the line, we can use the fact that two lines are perpendicular if the product of their slopes is -1. We know the slope of y=2x+3 is 2, so the slope of the line we are looking for is -1/2.

So, the equation of the line is y = -1/2x + b. To find the y-intercept, we can substitute the point (1,3) into the equation and solve for b.

y = -1/2x + b

3 = -1/2*1 + b

b = 3 + 1/2

b = 7/2

So, the equation of the line is y = -1/2x + 7/2

We can also write the equation of the line in the form ax+by=c where a, b and c are integers. To do this, we can multiply the equation by 2, to get rid of the fraction.

2y = -x + 7

2y + x = 7

So the equation of the line in the form ax+by=c is 2y+x=7

This line is the equation of the line that is perpendicular to y=2x+3 and passes through the point (1,3)

SpiritInly SpiritInly · Answer 2 · 2023-01-15T01:10:57+01:00

Answer:

2y+x=7

Step-by-step explanation:

The equation of a line is in the form:

[tex]y = mx + c[/tex]

where m is the gradient and c is the y-intercept.

In the equation y=2x+3 the gradient is 2, so to find the perpendicular gradient, it is the negative reciprocal, or in other words, flip the number upside down and make it negative:

[tex]2 = \frac{2}{1} [/tex]

[tex] \frac{2}{1} \: flipped \: upside \: down = \frac{1}{2} [/tex]

[tex] make \: \frac{1}{2} \: negative = - \frac{1}{2} [/tex]

So the gradient for the point (1, 3) is -1/2

Using y=mx+c to find c (the y-intercept) where y=1, m=-1/2, x=1, substitute these values in:

[tex]y = mx + c[/tex]

[tex]3 = - \frac{1}{2} (1) + c[/tex]

[tex]3 = - \frac{1}{2} + c[/tex]

[tex]3 + \frac{1}{2} = c[/tex]

[tex]c = \frac{7}{2} [/tex]

So the equation is:

[tex]y = - \frac{1}{2} x + \frac{7}{2} [/tex]

We need to write this in the form ax+by=c, so multiply everything by 2 to get rid of the fractions on the right:

[tex]2y = - x + 7[/tex]

Bring x to the left side by adding it:

[tex]2y + x = 7[/tex]

This is now in the form ax+by=c