Question

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

Short answer

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

Step-by-step solution

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 \(5 - 6 = -1\).

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.

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.