Constructing the Linear Equation from Two Points:

Given Points: A(2, 3) and B(4, 5)

1. Calculate the Slope (m):

m = (y₂ – y₁) / (x₂ – x₁) = (5 – 3) / (4 – 2) = 2 / 2 = 1

2. Use Point-Slope Form with one of the points (e.g., A):

y – y₁ = m(x – x₁)

y – 3 = 1(x – 2)

3. Simplify to Slope-Intercept Form (y = mx + b):

y = x + 1

The linear equation that passes through the points A(2, 3) and B(4, 5) is y = x + 1.

Constructing the Linear Equation from Two Points with Decimal Numbers:

Given Points: A(1.5, 2.3) and B(3.5, 4.1)

Step 1: Calculate the Slope (m)

Slope Formula: m = (y₂ – y₁) / (x₂ – x₁)

Calculating: m = (4.1 – 2.3) / (3.5 – 1.5) = 1.8 / 2 = 0.9

Step 2: Use Point-Slope Form with Point A

Point-Slope Formula: y – y₁ = m(x – x₁)

Identifying Parts: y – 2.3 = 0.9(x – 1.5) (Substitute y₁ = 2.3, x₁ = 1.5, and m = 0.9)

Step 3: Distribute the Slope (m)

Distributing: y – 2.3 = 0.9x – 1.35 (Distribute 0.9 to x and y₁)

Step 4: Move 2.3 to the Other Side

Moving: y = 0.9x – 1.35 + 2.3 (Add 2.3 to both sides)

Step 5: Simplify the Equation

Simplifying: y = 0.9x + 0.95 (Combine constants)

The linear equation passing through A(1.5, 2.3) and B(3.5, 4.1) is y = 0.9x + 0.95.

Constructing the Linear Equation from Two Points with Fractional Numbers:

Given Points: A\( \left(\frac{1}{2}, \frac{3}{4}\right) \) and B\( \left(\frac{3}{2}, \frac{5}{4}\right) \)

Step 1: Calculate the Slope (m):

Subtract the y-values: \( y₂ – y₁ = \frac{5}{4} – \frac{3}{4} = \frac{2}{4} = \frac{1}{2} \)

Subtract the x-values: \( x₂ – x₁ = \frac{3}{2} – \frac{1}{2} = \frac{2}{2} = 1 \)

Divide the differences: \( m = \frac{y₂ – y₁}{x₂ – x₁} = \frac{\frac{1}{2}}{1} = \frac{1}{2} \)

Step 2: Use Point-Slope Form with one of the points (e.g., A):

Insert the slope and one point into the formula: \( y – y₁ = m(x – x₁) \)

Substitute the values: \( y – \frac{3}{4} = \frac{1}{2}(x – \frac{1}{2}) \)

Step 3: Simplify to Slope-Intercept Form (y = mx + b):

Distribute the slope: \( y – \frac{3}{4} = \frac{1}{2}x – \frac{1}{4} \)

Add the y-intercept to both sides: \( y = \frac{1}{2}x + \frac{1}{2} \)

Conclusion: The linear equation that passes through the points A\( \left(\frac{1}{2}, \frac{3}{4}\right) \) and B\( \left(\frac{3}{2}, \frac{5}{4}\right) \) is \( y = \frac{1}{2}x + \frac{1}{2} \).