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In linear functions, the most popular way to write the equation is in the slope-intercept form. The slope-intercept form of a linear equation is given by \( y = mx + c \). Here, m is the slope of the line, and c is the y-intercept, or where the line crosses the y-axis.
This form is very useful because it allows us to see the slope and y-intercept directly from the equation. This makes graphs easier to understand.
To use the slope-intercept form effectively, first identify the slope, m, which tells us how steep the line is. A positive value of m means the line rises as it moves from left to right, while a negative value indicates it falls. In our example, the slope is \( -\frac{1}{2} \), meaning the line falls.
Next, the y-intercept, c, tells us where the line crosses the y-axis. Even if the line is steep or shallow, it has to cross the y-axis somewhere, and that's where c comes in handy. This is especially useful in practical applications like predicting values.

To find the y-intercept (c) in a linear function, we use known points and the slope of the line.
Given the slope, m, and a point (x, y) on the line, we can substitute them into the slope-intercept form equation and solve for c. Let's take a look at the original problem, where the slope m is \( -\frac{1}{2} \) and a given point (3, 7). By substituting these values into the slope-intercept form:

• Step 1: Start with the equation: \( y = -\frac{1}{2}x + c \)


• Step 2: Substituting the point (3, 7): \( 7 = -\frac{1}{2}(3) + c \)


• Step 3: Simplify to find c: \( 7 = -\frac{3}{2} + c \)


• Step 4: Add \( \frac{3}{2} \) to both sides: \(7 + \frac{3}{2} = c \)

After simplifying, we find that c = \frac{17}{2}. Now, we know that the y-intercept is \( \frac{17}{2} \), and it tells us the exact point where the line intersects the y-axis.

The substitution method is a straightforward technique to find unknowns in linear equations by substituting known values. It's especially helpful in the context of finding the y-intercept.
To use this method, we need a few pieces of information: the slope of the line ( m ) and at least one point ( x, y ) through which the line passes.
Using the original problem as an example, we know that m = -\frac{1}{2} and the point (3, 7) is on the line. We substitute these values into the slope-intercept equation to find c.
Here’s how we do it step-by-step:

• Step 1: Start with the slope-intercept formula: \( y = mx + c \)


• Step 2: Substitute the slope: \( y = -\frac{1}{2}x + c \)


• Step 3: Use the point (3, 7) to substitute x and y: \( 7 = -\frac{1}{2}(3) + c \)


• Step 4: Solve for c by isolating it on one side: \( 7 = -\frac{3}{2} + c \)


• Step 5: Add \( \frac{3}{2} \) to both sides: \( 7 + \frac{3}{2} = c \)


• Step 6: Simplify the equation to find \( c = \frac{17}{2} \)

Using substitution, we have successfully found the y-intercept (c) for our line.
This method is highly versatile and can be used in various types of linear equations to find unknown values.