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The slope-intercept form is a way of expressing linear equations. This form is extremely useful for graphing lines quickly because it directly shows the slope and the y-intercept of a line. The general expression for the slope-intercept form is \(y = mx + b\). Here, \(m\) represents the slope of the line, which indicates the steepness and direction, while \(b\) represents the y-intercept, the point where the line crosses the y-axis.
The conversion of a linear equation into slope-intercept form involves solving for \(y\) in terms of \(x\). Let's look at the example from the exercise:

• For the equation \(2x + 5y = 16\), we want to solve for \(y\): \[y = -\frac{2}{5}x + \frac{16}{5}\]
• For the equation \(3x - 7y = 24\), solving for \(y\) gives: \[y = \frac{3}{7}x - \frac{24}{7}\]

Having these equations in slope-intercept form makes it straightforward to find slopes and y-intercepts, setting the stage for graphing.

The point of intersection between two lines is where they cross each other on a graph. It is an essential concept when solving systems of linear equations since it represents a solution that satisfies both equations simultaneously.
To find this point, you equate the two equations if they are in slope-intercept form. In the provided solution, the equations \(y = -\frac{2}{5}x + \frac{16}{5}\) and \(y = \frac{3}{7}x - \frac{24}{7}\) are set equal. Solving the resulting equation allows you to find the x-coordinate of the intersection point.
Here are the steps performed:

• Set \(-\frac{2}{5}x + \frac{16}{5} = \frac{3}{7}x - \frac{24}{7}\) and multiply through by 35 to clear fractions.
• Solve the resulting equation for \(x\).
• Substitute \(x = 8\) back into one of the original equations to find \(y = 0\).

The point of intersection concluded from these steps is \((8, 0)\). This means both lines intersect at (8, 0) on the graph.

Graphing lines is a visual representation of equations in the Cartesian coordinates. To graph a line, knowing the slope and y-intercept is crucial, especially when lines are in the slope-intercept form.
To start graphing:

• Identify the y-intercept from the slope-intercept form \(y = mx + b\). The y-intercept is the point \((0, b)\).
• Use the slope \(m\) to determine the rise over the run. For example, a slope of \(-\frac{2}{5}\) means the line falls 2 units for every 5 units it runs to the right.
• Plot the y-intercept first, then use the slope to find another point.

Using these points, you can draw a line. In this exercise, the equations provided give us specific y-intercepts and slopes:

• The line \(y = -\frac{2}{5}x + \frac{16}{5}\) has a y-intercept of \((0, \frac{16}{5})\) and a slope of \(-\frac{2}{5}\).
• The line \(y = \frac{3}{7}x - \frac{24}{7}\) has a y-intercept of \((0, -\frac{24}{7})\) and a slope of \(\frac{3}{7}\).

These details help in accurately drawing lines and visually finding their intersection.

Linear equations describe relationships with constant rates of change. They are visually represented as straight lines on a graph. In general, linear equations take the form \(Ax + By = C\). Transforming them into the slope-intercept form makes it easier to solve graphically and analytically.
A linear equation features:

• Variables \(x\) and \(y\) raised to the first power only.
• Constant coefficients \(A\), \(B\), and \(C\).
• Operations including addition and multiplication.

The provided equations \(2x + 5y = 16\) and \(3x - 7y = 24\) are classic examples of linear equations. Solving problems involving linear equations often involves finding where these equations intersect, which indicates a common solution that satisfies both equations. These points of intersection can be graphically shown or algebraically solved as demonstrated in this solution.