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When dealing with a linear equation like \( y = -\frac{5}{6}x + 8 \), completing a table of solutions helps us understand how the value of \( y \) changes with different \( x \) values. For this equation, we calculated \( y \) values for \ x = -6 \, \ x = 0 \, and \ x = 6 \. We achieved this by substituting each \ x \ value into the equation and solving for \ y \. Here's a summary of the process:

• When \ x = -6 \, \ y = 13 \.
• When \ x = 0 \, \ y = 8 \.
• When \ x = 6 \, \ y = 3 \.

By completing this table, we get pairs of \( x, y \) values that ultimately help us graph the equation. The points we can plot are \( (-6, 13) \, (0, 8) \, and \ (6, 3) \). This approach makes the graphing process much more straightforward and verifies that the line passes through each point correctly.

Graphing a linear equation involves plotting points on a coordinate plane and drawing a line through those points. Using our equation \( y = -\frac{5}{6}x + 8 \), and the points from our table of solutions, we:

• Plot the point \ (-6, 13) \ on the coordinate plane.
• Plot the point \ (0, 8) \ on the coordinate plane.
• Plot the point \ (6, 3) \ on the coordinate plane.

After plotting these points, draw a straight line through each one. Linear equations always form a straight line, regardless of the values of \( x \) and \( y \). The points confirm our equation is correctly graphed when the line passes through them perfectly. This method helps visually confirm the equation's solutions and better understand the relationship between \( x \) and \( y \).

The equation \( y = -\frac{5}{6}x + 8 \) is in slope-intercept form, a common way to express linear equations. In slope-intercept form, an equation appears as \ y = mx + b \, where:

• \ m \ is the slope of the line, which represents the rate of change. In our equation, \ m = -\frac{5}{6} \, indicating the line slopes downward at a ratio of \ -5 \ to \ 6 \.
• \ b \ is the y-intercept, the point where the line crosses the y-axis. For our equation, \ b = 8 \ means the line crosses the y-axis at \ y = 8 \.

The slope tells us the steepness and direction of the line. A negative slope means the line decreases as \ x \ increases, moving from left to right. The y-intercept is helpful for starting the graph, offering a point where \ x = 0 \ and then use the slope to find other points. Understanding both components is crucial for graphing and interpreting linear equations, providing a visual way to understand algebraic relationships.