91Ó°ÊÓ

Solving linear equations is all about finding the value of the variable that makes the equation true. In our exercise, we start with the equation: \( y - 3 = \frac{1}{2}(x - 4) \). When dealing with fractions, it's often easier to clear them first. We do this by multiplying every term by the denominator. Here, we multiply both sides by 2 to get rid of the fraction, resulting in:
\[ 2(y - 3) = 2 \cdot \frac{1}{2}(x - 4) \] This simplifies to:
\[ 2y - 6 = x - 4 \] Next, we need to isolate the variable \( y \) to put the equation into what we call the slope-intercept form, which looks like this: \( y = mx + b \). Adding 6 to both sides, we get:
\[ 2y = x + 2 \] Finally, dividing everything by 2, we arrive at:
\[ y = \frac{1}{2}x + 1 \] This equation is now in slope-intercept form, which makes it easier to identify the slope and the intercepts.

Graphing a linear equation involves plotting points that the equation passes through and then drawing the line that connects them. The slope-intercept form \( y = mx + b \), like the one we found: \( y = \frac{1}{2}x + 1 \), is particularly useful. Here's why:

• The coefficient of \( x \) is the slope (\( m \)).
• The constant term (\( b \)) is the y-intercept.

To graph this equation:

• Start at the y-intercept, which is the point \( (0, 1) \). This is where the line crosses the y-axis.
• Use the slope to find another point. The slope \( \frac{1}{2} \) means that for every 2 units you move to the right, you move 1 unit up. Starting from \( (0, 1) \), if you go right 2 units to \( (2, 1) \), then move up 1 unit to \( (2, 2) \), you have another point.

Connect these points with a straight line, and you've graphed the equation! Remember, linear equations form straight lines when graphed.

Intercepts are points where the graph of an equation crosses the axes. There are two kinds:

• Y-intercept: The point where the graph crosses the y-axis. This happens when \( x = 0 \). From our equation \( y = \frac{1}{2}x + 1 \), substituting \( x = 0 \) gives \( y = 1 \). So, the y-intercept is \( (0, 1) \).
• X-intercept: The point where the graph crosses the x-axis. This happens when \( y = 0 \). To find this, set \( y = 0 \) in \( y = \frac{1}{2}x + 1 \):
  \[ 0 = \frac{1}{2}x + 1 \] Solve for \( x \):
  \[ -1 = \frac{1}{2}x \] Multiply both sides by 2:
  \[ x = -2 \] So, the x-intercept is \( (-2, 0) \).

Identifying intercepts is key to quickly sketching the graph and understanding the behavior of linear equations.