91Ó°ÊÓ

Parallel lines are lines in a plane that never meet. They have the same slope but can have different y-intercepts. For instance, if one line has the equation \(y = mx + b\) and another line has a different equation \(y = mx + c\), they will be parallel as long as the value of \(m\) (the slope) is the same.

In the given exercise, you are given a line described by \(x + y = 7\). To determine if another line is parallel to it, we first rewrite this equation in slope-intercept form (\(y = mx + b\)) so we can clearly see the slope. Converting \(x + y = 7\) to \(y = -x + 7\), we see that the slope \(m\) is \(-1\). Therefore, any line with a slope of \(-1\) will be parallel to the original line.

Understanding this is crucial for solving problems involving parallel lines.

The slope-intercept form of a line's equation is one of the most common ways to write the equation of a straight line. It is written as \(y = mx + b\), where \(m\) represents the slope of the line and \(b\) represents the y-intercept — the point where the line crosses the y-axis.

Knowing how to convert an equation to this form is very useful because it directly shows you the slope and y-intercept.

• The slope \(m\) indicates the steepness of the line: how much \(y\) changes for a unit change in \(x\).
• The y-intercept \(b\) tells you the value of \(y\) when \(x\) is zero.

In our exercise example, the line \(x + y = 7\) was converted to slope-intercept form as \(y = -x + 7\). The slope is \(-1\), confirming that any parallel line will also have a slope of \(-1\).

The point-slope form is another essential method to express the equation of a straight line. It is particularly useful when you know the slope of the line and one point it passes through. The general form is \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is a point on the line and \(m\) is the slope.

To use the point-slope form:

• Identify the slope \(m\).
• Use the coordinates of the given point \((x_1, y_1)\).
• Substitute these values into the point-slope equation.

In our exercise, the goal was to find the equation of a line parallel to \(x + y = 7\) and passing through the point \((-3, 2)\). We already determined that the slope \(m\) is \(-1\). Using the point-slope form:
\(y - 2 = -1(x + 3)\)
Simplifying this equation:

• Distribute the slope:\(y - 2 = -x - 3\).
• Isolate \(y\):\(y = -x - 1\).

This gives us the final equation of the required line.