Graphing linear equations

Question

A line passes through A(3,7), and B(-4,9). Find the value of "a" if C(a,1) is on the line. 

 

I don't understand what I need to do to solve this problem.

Answer 1

Given two points A(3,7) and B(-4,9) .

 

• Use the point-slope form equation to solve for the slope (m).
• y - y₁ = m ( x - x₁₎ Equation 1.
  

1. Let x = 3, X₁ = -4
2. Let y = 7, y₁ = 9  Substitute these values in to Equation 1.
3. (7 - 9) = m ( 3 - (-4))
4. -2 = m (7)
5. m = -2/7

• Now use the slope-intercept form equation to solve for the value of (b). 
• y = mx + b   Equation 2.

1. Use point A to solve for (b).
2. (7) = (-2/7)(3) + b
3. 7 = -6/7 + b
4. 7 + 6/7 = b
5. b = 55/7

Therefore the equation for the line with points A and B is given by:

• y = -2/7x + 55/7
• Now substitute the point (a,1) into the equation and solve for (a).
• (1) = -2/7(a) + 55/7
• 1 = -2/7a + 55/7
• 1 - 55/7 = -2/7a. The left hand side of the equation is equal to (7-55)/7 = -48/7.
• -48/7 = -2/7a
• (-48/7)/(-2/7) = a
• 24 = a      Answer...

 

Answer 2

I was going on the same route as James and realized something.

 

As he mentioned, m = -2/7. In order for our 3rd point to be on the line, the slope has to be the same! We can pick either point...

 

(7-1)/(3-a) = -2/7

6/(3-a) = -2/7

 

Cross multiplying, -2(3-a) = 42

-6+2a = 42

2a = 48

a = 24

 

Boom.

 

 

Answer 3

Two points determine a line.  The problem gives point A(3,7) and point B(-4,9).  It also says that point C(a,1) is on the line.

 

Now, 'a' is the value of x when y=1.  That is, C(a,1) is on the line, so x=a and y=1.  We can determine the value of 'a' by first determining the equation of the line.

 

   y-y₁ = ( (y₂-y₁) / (x₂-x₁) ) (x - x₁)

   y-7 =  ( (9-7)) / ((-4)-3) ) (x - 3)

   y-7 = (-2/7)x + 6/7

   y = (-2/7)x + 55/7

 

Now, solve for x (that is, a) when y=1.

   1 = (-2/7)a + 55/7

   (2/7)a = 48/7

      2a = 48

       a = 24

The value of 'a' is 24.  The point (24,1) is on the line.

 

Check:

  Is 1 = (-2/7)(24) + 55/7   ?

      7 = -48 + 55   ?

      7 = 7   ?yes

 

Since you mentioned 'graphing linear equations,' when you plot the line, the point C(24,1) should be on the line.

Answer 4

 

First, find the slope of the line using the formula 

 

   m = (y2 - y1)/(x2 - x1)

 

        x1 = 3

        y1 = 7

        x2 = -4

        y2 = 9

 

   m = (9 - 7)/(-4-3)

       = 2/(-7)

       = -2/7

 

Use slope intercept form, and substitute the slope and one of the given points ... solve for b

 

    y = mx + b

    7 = (-2/7)(3) + b

    7 = -6/7 + b

    

    b = 6/7 + 7

    b = 6/7 + 49/7

    b = 55/7

 

The equation of the line is y = -(2/7)x + 55/7

 

Since (a, 1) is on the line, we can plug that point into the equation, and solve for a

 

    1 = -(2/7)(a) + 55/7

    1 = -2a/7 + 55/7

    7 = -2a + 55

    -48 = -2a

       a = 24