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Linear equations are mathematical statements that describe a straight line when graphed on a coordinate plane. They take the form of \( y = mx + b \). Here, \( m \) represents the slope, showing how steep the line is, and \( b \) represents the y-intercept, indicating where the line crosses the y-axis.

In the given exercise, both equations are linear:

• First equation: \( y = \frac{1}{2} x + 4 \)
• Second equation: \( y = \frac{1}{2} x - 1 \)

Both equations share the same slope \( \frac{1}{2} \), meaning the lines they represent are parallel and have the same steepness. Understanding this helps predict that any intersection will reveal where these lines cross.

Linear equations are crucial in algebra and various applications, from physics to economics. One practical way to solve them, as shown, is by graphing. The visual method helps us see solutions directly.

The intersection point is where two lines cross on a graph. This point has coordinates \((x, y)\) that satisfy both equations in a system.

In our exercise, we plot the lines for the equations \( y = \frac{1}{2} x + 4 \) and \( y = \frac{1}{2} x - 1 \). Observing where these lines intersect tells us the solution.

• Step 1: Plot \( y = \frac{1}{2} x + 4 \). Note that the y-intercept is 4 and the slope is \( \frac{1}{2} \).
• Step 2: Plot \( y = \frac{1}{2} x - 1 \), with a y-intercept at -1 and the same slope.
• Step 3: Identify the intersection point by seeing where the two lines meet.

Once plotted, these steps reveal the intersection point at (10, 9). This is because both lines share the same slope but different y-intercepts, making them intersect at exactly one point. This point, (10, 9), solves both linear equations simultaneously.

The y-intercept is where a line crosses the y-axis (vertical axis) on a graph. It occurs where \( x = 0 \). In the linear equation format \( y = mx + b \), \( b \) is the y-intercept. It's a starting point when plotting a line.

For our given equations:

• In \( y = \frac{1}{2} x + 4 \), the y-intercept is 4. So, when \( x = 0 \), \( y = 4 \).
• In \( y = \frac{1}{2} x - 1 \), the y-intercept is -1. Thus, when \( x = 0 \), \( y = -1 \).

Identifying y-intercepts makes graphing easier. Begin by plotting these points on the y-axis.

From these points, use the slope to find other points. The slope tells us how much \( y \) increases or decreases for a given \( x \). These y-intercepts guide the graphing process and help visually solve the system of equations.