Question

True or False. The point (-2,-6) is on the graph of the equation \(x=2 y-2\).

Short answer

False

Step-by-step solution

Understand the equation and point

First, recall that the given equation is in the form of a linear equation: \(x = 2y - 2\). And the point given is (-2, -6). We need to determine if this point satisfies the equation.

Substitute the coordinates into the equation

Substitute the coordinates of the point (-2, -6) into the equation: \(x = 2y - 2\)This means substitute \(x = -2\) and \(y = -6\).

Perform substitution

Replace \(x\) with -2 and \(y\) with -6 in the equation:\(-2 = 2(-6) - 2\)

Simplify the equation

Simplify the right-hand side of the equation:\(-2 = -12 - 2\)\(-2 = -14\)

Compare both sides

Observe that the left side, \(-2\), is not equal to the right side, \(-14\). Therefore, the initial equation is not satisfied with the point (-2, -6).

Conclusion

Since the left side does not equal the right side after substitution, the point (-2, -6) does not lie on the graph of the equation \(x = 2y - 2\). Therefore, the statement is False.

Key concepts

Point Substitution

Point substitution is a method used to determine if a particular point lies on the graph of a given equation. It involves replacing the variables in the equation with the coordinates of the point in question. Specifically, for an equation with variables x and y, you substitute the x-coordinate and the y-coordinate of the point into the respective variables in the equation. This method essentially checks if the ordered pair satisfies the equation. For instance, if we have the equation \(x = 2y - 2\) and the point (-2, -6), we substitute \(-2\) for x and \(-6\) for y into the equation to see if both sides of the equation remain equal. When performed correctly, point substitution helps in verifying whether a point lies on the curve represented by the equation.

Graph of an Equation

A graph of an equation is a visual representation of all the solutions of that equation. Each solution pair \( x, y \) when plotted on the coordinate axes gives a point on the graph. For linear equations like \( x = 2y - 2 \), the graph will be a straight line. To draw the graph, you can follow these steps:

• Choose several values for y and calculate the corresponding x values using the equation.
• Plot the resulting points (x, y) on the coordinate plane.
• Connect the points to form a straight line.

This visual representation helps in understanding the relationship between x and y. Furthermore, it allows us to easily identify whether a specific point lies on the line by checking if it visually aligns with the plotted points. In our original exercise, the point (-2, -6) does not lie on the graph of the equation \(x = 2y - 2\), which we can confirm both algebraically (through substitution) and visually (on the graph).

Equation Verification

Equation verification is the process of checking whether a given point satisfies the equation of a graph. This involves substituting the point’s coordinates into the equation and simplifying to see if both sides of the equation are equal. If they are, the point is on the graph; if not, it is not. For our specific problem:

• We substituted \(x = -2\) and \(y = -6\) into \(x = 2y - 2\).
• This gave us the equation \(-2 = 2(-6) - 2\).
• Upon simplifying, we got \(-2 = -12 - 2\) which further simplifies to \(-2 = -14\).

Since \(-2\) is not equal to \(-14\), the point (-2, -6) does not satisfy the equation \(x = 2y - 2\). Thus, the statement that it lies on the graph is false. Following these detailed steps and understanding each process helps in verifying the accuracy of any point relative to a given equation.