Problem 56 - Algebra 1

Question

Decide whether the given point lies on the line. Justify your answer both algebraically and graphically. $$5 x+6 y=-1 ;(1,-1)$$

Step-by-Step Solution

Answer

The point (1, -1) lies on the line 5x + 6y = -1 as it satisfies the equation algebraically as well as graphically.

Step 1: Algebraic Verification

Substitute the given point (1, -1) into the equation of the line (5x + 6y = -1). So place \(1\) for \(x\) and \(-1\) for \(y\). After substitution the equation becomes \(5*1 + 6*(-1) = -1\), which simplifies to \(5 - 6 = -1\).

Step 2: Verify The Equation

Now solve the equation \(5 - 6 = -1\). This simplifies to \(-1 = -1\). Both sides of the equation match, confirming that the point (1, -1) indeed lies on the line mathematically.

Step 3: Graphical Verification

To confirm graphically, plot the line using its equation and then mark the point on the graph. The equation can be rewritten as \(y = -\frac{5}{6}x - \frac{1}{6}\) to easily identify the slope and y-intercept. Upon plotting the point (1,-1), it would be visible that the point lies on the line.

Key Concepts

Algebraic Verification

To determine if a specific point such as \((1, -1)\) lies on a line described by a linear equation like \(5x + 6y = -1\), we use algebraic verification. This involves substituting the coordinates of the point into the equation. In this example, plug \(x = 1\) and \(y = -1\) into the equation:

• Start with the given point \((1, -1)\).
• Substitute \(1\) for \(x\) and \(-1\) for \(y\).
• The equation becomes \(5(1) + 6(-1) = -1\).
• Solve it: \(5 - 6 = -1\), which simplifies to \(-1 = -1\).

If both sides of the equation are equal, the point indeed lies on the line. Here, the equation balances, confirming \((1, -1)\) is on the line algebraically.

Graphical Verification

Graphical verification provides a visual way to confirm if a point is on a line. To achieve this, you can plot the equation and the point on a graph. Let's break it down using the line equation \(5x + 6y = -1\).

• First, rearrange the equation to slope-intercept form: \(y = -\frac{5}{6}x - \frac{1}{6}\).
• You can easily identify the slope \(-\frac{5}{6}\) and y-intercept \(-\frac{1}{6}\).
• Draw the line using the slope and y-intercept.
• Plot the given point \((1, -1)\) on this same graph.

If the point lies directly on the plotted line, it confirms visually that the point is indeed part of the line. This additional step supports the algebraic finding, ensuring accuracy.

Point on Line

Understanding whether a point \((x, y)\) lies on a line is crucial when dealing with linear equations. We use the expression of the line to verify this. A point on a line means that when its coordinates are plugged into the equation, the equation holds true. To dive deeper:

• A line in a plane is the set of all points that satisfies its equation.
• If substituting \(x\) and \(y\) in the equation fulfills the equation, the point is on the line.
• In algebraic terms, it means both sides of the equation equal after substitution.
• Graphically it implies the point lies on the plotted line path.

This understanding can be crucial for solving problems in geometry and calculus, where coordinates and lines are fundamental components.