Build slope-intercept form from two points

• Compute slope as vertical change divided by horizontal change with a consistent point order.
• Substitute a known point into y equals m x plus b to determine the intercept.
• Verify a line equation by checking that both original points satisfy it.

Build the idea

Worked example

1. Step 1

   Subtract y-coordinates and x-coordinates in the same order. The rise is eight and the run is four, so the constant rate of change is two.

   $$m=\frac{8}{4}=2$$

2. Step 2

   Insert x equals two, y equals three, and m equals two into slope-intercept form. Solving three equals four plus b gives b equals negative one.

   $$3=2(2)+b\Rightarrow b=-1$$

3. Step 3

   Combine the slope and intercept in one equation. The coefficient two describes the rise per unit run, while negative one is the y-value when x is zero.

   $$y=2x-1$$

• Subtracting x-coordinates in one order and y-coordinates in the opposite order.

  Label the points first and use the same second-minus-first order in both numerator and denominator; otherwise the slope sign is reversed.

• Treating the slope as the intercept and writing y equals two x plus two.

  Slope and intercept describe different features. Substitute a known point to calculate b instead of copying the slope into both positions.

Find the slope-intercept equation through negative one comma four and three comma negative four, then check both points.

This reviewed reading order covers the lesson explanation, equations, example, and checks without claiming that a video exists.

1. Part 1

   Lesson overview

   Two distinct points with different x-coordinates determine one nonvertical line. Their vertical change divided by horizontal change gives the slope, and either point then reveals the intercept. A line records a constant rate of change. Moving from one known point to another creates a horizontal change and a vertical change, and their ratio is the same for every nonzero horizontal move along that line. Compute slope as vertical change divided by horizontal change with a consistent point order.

2. Part 2

   Visual model and equations

   The order used for subtraction does not matter if it is consistent. Reversing both changes reverses both signs, so the quotient remains unchanged. Mixing the point order in only one subtraction creates an incorrect sign. Plot the points two comma three and six comma eleven. A horizontal run of four units and a vertical rise of eight units form a right triangle, so the line rises two units for every one unit moved right. The slope equals the quantity eleven minus three divided by the quantity six minus two, which equals two. y equals two x minus one.

   • The slope equals the quantity eleven minus three divided by the quantity six minus two, which equals two.
   • y equals two x minus one.
3. Part 3

   Worked example

   Find the equation of the line through the points two comma three and six comma eleven, then verify the equation using both coordinates. Calculate the slope from paired coordinate differences, substitute the first point to solve for b, and test the finished equation at both x-values. Subtract y-coordinates and x-coordinates in the same order. The rise is eight and the run is four, so the constant rate of change is two. Insert x equals two, y equals three, and m equals two into slope-intercept form. Solving three equals four plus b gives b equals negative one. Combine the slope and intercept in one equation. The coefficient two describes the rise per unit run, while negative one is the y-value when x is zero. The line through the two points is y equals two x minus one. At x equals two the formula gives three, and at x equals six it gives eleven. Both original points satisfy the same equation, so the slope and intercept are consistent.

   • m equals eight fourths, which equals two.
   • Three equals two times two plus b, so b equals negative one.
   • y equals two x minus one.
4. Part 4

   Checks, practice, and scope

   After the slope is known, the intercept is not guessed from a sketch. Substituting either point into slope-intercept form turns the unknown intercept into a one-step equation, and checking the second point tests the complete result. Subtracting x-coordinates in one order and y-coordinates in the opposite order. Label the points first and use the same second-minus-first order in both numerator and denominator; otherwise the slope sign is reversed. Treating the slope as the intercept and writing y equals two x plus two. Slope and intercept describe different features. Substitute a known point to calculate b instead of copying the slope into both positions. Find the slope-intercept equation through negative one comma four and three comma negative four, then check both points. The vertical change is negative eight while the horizontal change is four; use one point to determine b. The slope is negative two and the intercept is two, so the equation is y equals negative two x plus two. Consistent coordinate subtraction preserves the correct slope sign. A known point converts the unknown intercept into a solvable equation. Checking both points verifies the complete line rather than only one parameter. This method assumes the two points have different x-coordinates; a vertical line has undefined slope and cannot use y equals m x plus b. The lesson addresses exact coordinates and does not estimate a best-fit line from noisy statistical data.

Scope and limitations

• This method assumes the two points have different x-coordinates; a vertical line has undefined slope and cannot use y equals m x plus b.
• The lesson addresses exact coordinates and does not estimate a best-fit line from noisy statistical data.