Problem 10 - Understanding Elementary Algebra with Geometry

Question

Find the \(x\) - and \(y\) -intercepts of the equation. $$x+y=9$$

Step-by-Step Solution

Answer

The x-intercept is (9, 0) and the y-intercept is (0, 9).

Step 1: Identify the equation

The given equation is \(x + y = 9\).

Step 2: Find the x-intercept

To find the x-intercept, set \(y = 0\) and solve for \(x\). Substitute \(y = 0\) into the equation: \(x + 0 = 9\), so \(x = 9\). The x-intercept is \((9, 0)\).

Step 3: Find the y-intercept

To find the y-intercept, set \(x = 0\) and solve for \(y\). Substitute \(x = 0\) into the equation: \(0 + y = 9\), so \(y = 9\). The y-intercept is \((0, 9)\).

Key Concepts

x-intercept

The x-intercept is the point where a graph crosses the x-axis. At this point, the value of y is always zero because the point lies on the x-axis.
To find the x-intercept of the equation given, we set y to 0 and solve for x.

For the equation given:
\(x + y = 9\)
Set \(y = 0\):
\(x + 0 = 9\)
Simplify to find:
\(x = 9\)
This gives us the x-intercept: \( (9, 0) \).
So, anytime you need to find the x-intercept of a linear equation, you set y to 0 and solve for x. This is a crucial concept in graphing linear equations!

y-intercept

The y-intercept is the point where a graph crosses the y-axis. At this point, the value of x is always zero because the point lies on the y-axis.
To find the y-intercept of the equation given, we set x to 0 and solve for y.

For the equation given:
\(x + y = 9\)
Set \(x = 0\):
\(0 + y = 9\)
Simplify to find:
\(y = 9\)
This gives us the y-intercept: \( (0, 9) \).
So, to find the y-intercept of any linear equation, you just set x to 0 and solve for y. This is very useful when plotting or analyzing graphs.

linear equations

Linear equations are equations of the first order. This means they have no exponents higher than one and graph as straight lines. The general form of a linear equation is
\[ax + by = c \],
where a, b, and c are constants.

• One main feature is that the graph of a linear equation is always a straight line.
• They can be written in different forms like slope-intercept form: \[y = mx + b\]

Finding the x- and y-intercepts gives us specific points on the graph which help in sketching the line.
Understanding linear equations is key for many areas of algebra and helps build the foundation for more complex topics.