How do you write an equation in slope intercept form for the line through the given points (7,5 ); (-1, 1/5)?

Question

Nathan Axel · Accepted Answer

#y = color(red)(3/5)x + color(blue)(4/5)#

First, we need to determine the slope. The slope can be found by using the formula: #m = (color(red)(y_2) - color(blue)(y_1))/(color(red)(x_2) - color(blue)(x_1))#

Where #m# is the slope and (#color(blue)(x_1, y_1)#) and (#color(red)(x_2, y_2)#) are the two points on the line.

Substituting the values from the points in the problem gives:

#m = (color(red)(1/5) - color(blue)(5))/(color(red)(-1) - color(blue)(7)) = (color(red)(1/5) - (5/5 xx color(blue)(5)))/(color(red)(-1) - color(blue)(7)) = (color(red)(1/5) - 25/5)/(color(red)(-1) - color(blue)(7))#

#m = (-24/5)/-8 = 24/40 = (8 xx 3)/(8 xx 5) = (color(red)(cancel(color(black)(8))) xx 3)/(color(red)(cancel(color(black)(8))) xx 5) = 3/5#

Now we can use the point-slope formula to write an equation for the line. The point-slope formula states: #(y - color(red)(y_1)) = color(blue)(m)(x - color(red)(x_1))#

Where #color(blue)(m)# is the slope and #color(red)(((x_1, y_1)))# is a point the line passes through.

Substituting the slope we calculated and the first point from the problem gives:

#(y - color(red)(5)) = color(blue)(3/5)(x - color(red)(7))#

The slope-intercept form of a linear equation is: #y = color(red)(m)x + color(blue)(b)#

Where #color(red)(m)# is the slope and #color(blue)(b)# is the y-intercept value. Solving the equation we found for #y# gives:

#y - color(red)(5) = (color(blue)(3/5) xx x) - (color(blue)(3/5) xx color(red)(7))#

#y - color(red)(5) = 3/5x - 21/5#

#y - color(red)(5) + 5 = 3/5x - 21/5 + 5#

#y - 0 = 3/5x - 21/5 + (5 xx 5/5)#

#y = 3/5x - 21/5 + 25/5#

#y = color(red)(3/5)x + color(blue)(4/5)#

Avery Alexander · Answer

#y=3/5x+4/5#

The equation of a line in #color(blue)"slope-intercept form"# is.

#color(red)(bar(ul(|color(white)(2/2)color(black)(y=mx+b)color(white)(2/2)|)))# where m represents the slope and b, the y-intercept.

To calculate m, use the #color(blue)"gradient formula"#

#color(red)(bar(ul(|color(white)(2/2)color(black)(m=(y_2-y_1)/(x_2-x_1))color(white)(2/2)|)))# where # (x_1,y_1),(x_2,y_2)" are 2 coordinate points"#

The 2 coordinate points are # (7,5)" and (-1,1/5)#

let # (x_1,y_1)=(-1,1/5)" and " (x_2,y_2)=(7,5)#

#rArrm=(5-1/5)/(7+1)=(24/5)/8=3/5#

We can write the partial equation as #y=3/5x+b#

To find b, substitute either of the 2 given points into the partial equation and solve for b.

#"Using " (7,5)" that is " x=7,y=5#

#rArr5=(3/5xx7)+b#

#rArrb=5-21/5=25/25-21/25=4/5#

#rArry=3/5x+4/5" is equation in slope-intercept form"#

Avery Alexander · Answer

undefined To write an equation in slope-intercept form for the line passing through the points (7,5) and (-1, 1/5), follow these steps:

1. Find the slope (m) using the formula:
   m = (y₂ - y₁) / (x₂ - x₁)

2. Substitute the coordinates of the points into the formula:
   m = (1/5 - 5) / (-1 - 7)

3. Simplify the expression:
   m = (1/5 - 5) / (-1 - 7) = (-24/5) / (-8) = 3/5

4. Use one of the points and the slope in the point-slope form equation:
   y - y₁ = m(x - x₁)

5. Substitute the values of one of the points and the slope into the equation:
   y - 5 = (3/5)(x - 7)

6. Distribute the slope:
   y - 5 = (3/5)x - 21/5

7. Move the constant to the other side of the equation:
   y = (3/5)x - 21/5 + 5

8. Simplify:
   y = (3/5)x - 21/5 + 25/5

9. Combine like terms:
   y = (3/5)x + 4/5

10. Final equation in slope-intercept form:
   y = (3/5)x + 4/5