Problem 54 - Algebra 1

Question

Decide whether the given point lies on the line. Justify your answer both algebraically and graphically. $$x-y=10 ;(5,-5)$$

Step-by-Step Solution

Answer

Yes, the point (5,-5) lies on the line \( x - y = 10 \)

Step 1: Substitute the point into the line equation

The given line equation is \( x - y = 10 \). We can input the x value (5) and the y value (-5) from our point into the equation. This will give us \( 5 - (-5) = 10 \), which simplifies to \( 5 + 5 = 10 \), and thus \( 10 = 10 \).

Step 2: Verification

As the left side of the equation equals the right side, the equation holds true. Therefore, it verifies that the given point (5,-5) lies on the line.

Step 3: Graphical verification

To verify graphically, you should plot the line equation \( x - y = 10 \) and the point (5,-5) on a graph. If the point lies on the line, it confirms graphically that the point belongs to the line. As both algebraical and graphical confirmations align, we can be sure the point lies on the line.

Key Concepts

Algebraic Verification

To understand algebraic verification, let's consider a linear equation like \( x - y = 10 \). The task is to check if a point, in this case \((5, -5)\), lies on this line. This is done by substituting the point's coordinates \(x = 5\) and \(y = -5\) into the equation. The substitution results in \(5 - (-5) = 10\). Simplifying the equation, we get \(5 + 5 = 10\), which is true since \(10 = 10\). This means that the calculation confirms the point is on the line, because both sides of the equation balance. The algebraic method is straightforward:

• Substitute the x-coordinate of the point into the equation for \(x\).
• Substitute the y-coordinate of the point into the equation for \(y\).
• If the equation holds true after simplification, the point lies on the line.

Algebraic verification essentially proves that the coordinates satisfy the line equation, meaning the point is accurately positioned on the graph.

Graphical Verification

Graphical verification offers a visual understanding of why a point lies on a linear equation. In this method, you need to sketch the line given by the equation \(x - y = 10\) and then plot the point \((5, -5)\) on the same graph. If the point shares the same path as the line, it verifies that the point lies on it.To draw the line:

• Identify two easy points on the line by assigning values to \(x\) or \(y\) and solving for the other variable. For instance, when \(x = 10\), \(y = 0\) (as \(10 - 0 = 10\)); and when \(x = 0\), \(y = -10\) (as \(0 - (-10) = 10\)).
• Draw the line passing through these points.

After plotting the point \((5, -5)\), determine if it falls on this line. By seeing the point and line overlap, you visually confirm the point's presence on the line. Graphical verification is great for intuitive understanding and quick checks.

Coordinate Geometry

Coordinate geometry is the mathematical study of geometry using a coordinate system. This powerful tool connects algebra and geometry, offering ways to visualize mathematical concepts and solutions. For the equation \(x - y = 10\), coordinate geometry involves plotting the equation in a coordinate plane and examining the alignment of points and lines.This system helps in:

• Visualizing how changes in equations affect shapes and figures on the graph.
• Understanding relationships and distances between points.
• Determining the positions and interactions of lines on a plane.

In our example, coordinate geometry facilitates illustrating how the line \(x - y = 10\) and the point \((5, -5)\) relate. When we plot and see both elements coincide, it validates algebraic and graphical assertions. Using the plane as a canvas helps students "see" the answer, bolstering their spatial reasoning with coordinate geometry.