Write an equation in standard form for the line through the given points.

a)  A(4, 5) and B(3, 7)

To find the equation of a line passing through two points \( A(4, 5) \) and \( B(3, 7) \), follow these steps:

### 1. Find the slope of the line.
The formula for the slope \( m \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Substitute the coordinates of points \( A(4, 5) \) and \( B(3, 7) \):

\[
m = \frac{7 - 5}{3 - 4} = \frac{2}{-1} = -2
\]

### 2. Use the point-slope form to find the equation.
The point-slope form of the equation of a line is:

\[
y - y_1 = m(x - x_1)
\]

Substitute the slope \( m = -2 \) and one of the points, say \( A(4, 5) \):

\[
y - 5 = -2(x - 4)
\]

Simplify:

\[
y - 5 = -2x + 8
\]

\[
y = -2x + 13
\]

### 3. Convert to standard form.
The standard form of a linear equation is \( Ax + By = C \), where \( A \), \( B \), and \( C \) are integers, and \( A \geq 0 \).

Starting with \( y = -2x + 13 \), rearrange it to put it into standard form:

\[
2x + y = 13
\]

This is the equation of the line in standard form.

### Final Answer:
The equation of the line through points \( A(4, 5) \) and \( B(3, 7) \) in standard form is:

\[
2x + y = 13
\]

Would you like more details or have any questions? Here are five related questions to explore further:

1. How would the standard form change if one of the points was at the origin?
2. What if the slope was zero? What would the line's equation look like?
3. How do we find the y-intercept from the slope-intercept form?
4. Can you explain why we rearrange the equation to standard form?
5. How do you interpret the slope of \( -2 \) in the context of this line?

**Tip:** The slope tells you the rate at which the y-coordinate changes with respect to the x-coordinate.

Write an equation in standard form for the line through the given points: A(4, 5) and B(3, 7).

Learn how to write the equation of a line passing through points A(4,5) and B(3,7) in standard form. This solution involves key algebraic concepts like the slope formula and point-slope form, suitable for students in Grades 8-10.

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