Question

Decide whether the given point lies on the line. Justify your answer both algebraically and graphically. $$-4 x-3 y=-8 ;(-4,2)$$

Short answer

The point (-4, 2) does not lie on the line -4x -3y = -8 both algebraically and graphically.

Step-by-step solution

Algebraic Justification

Substitute the point (-4,2) into the equation, replacing x and y respectively: \(-4*-4 - 3*2 = -8 \). Evaluate both expressions to check if they equate: \(16 - 6 = 10\) which is not equal to -8. So algebraically, the point (-4, 2) does not lie on the line \(-4x -3y = -8\).

Graphical Justification

To graphically demonstrate this, start by graphing the equation of the line. First, rewrite the equation to the slope-intercept form, y = mx + b: \(-4x -3y = -8\)becomes \(y = -\(\frac{4}{3}\)x + \(\frac{8}{3}\). Graph the line and the point (-4, 2). By observation, it's noticeable that the point (-4, 2) does not lie on the line. Therefore, the point (-4,2) does not lie on the line graphically either.

Key concepts

Algebraic Justification

Determining whether a point is on a given line can be achieved through algebraic justification. This process involves substituting the x and y values of the point into the equation of the line. For instance, the exercise involves the equation \( -4x - 3y = -8 \) and asks us to justify if the point \( (-4,2) \) lies on it.

Here's how we do it: For the point \( (-4,2) \) the x-value is -4 and the y-value is 2. When these values are substituted into the line's equation, we get \( -4*(-4) - 3*2 \). Simplifying this results in \( 16 - 6 = 10 \), which is clearly not equal to -8. Thus, algebraically, we can say the point does not satisfy the line's equation, hence it does not lie on the line.

Graphical Justification

Graphical justification involves visual inspection of the line and the point on a coordinate system. This method can be useful especially when you're dealing with simple linear equations or when you want to get a visual sense of where the given point lies in relation to the line.

To graphically justify whether a point lies on a line, first convert the equation to slope-intercept form, which is \( y = mx + b \). For the given example, we manipulate \( -4x - 3y = -8 \) to get \( y = -\frac{4}{3}x + \frac{8}{3} \). Once the line is graphed using this equation, you can plot the point \( (-4,2) \) on the same graph. If the point does not sit on the line, as it is in our case, the graphical justification supports the earlier algebraic findings that the point \( (-4,2) \) does not lie on the given line.

Slope-Intercept Form

Understanding the slope-intercept form of a linear equation is fundamental in algebra. Expressed as \( y = mx + b \) where \( m \) is the slope and \( b \) is the y-intercept, this format allows for quick graphing and analysis of lines. The slope, \( m \), represents the line's steepness and direction, while the y-intercept, \( b \) indicates where the line crosses the y-axis.

In our exercise, converting the given line equation to slope-intercept form revealed the slope, \( -\frac{4}{3} \) and the y-intercept, \( \frac{8}{3} \). A negative slope means the line goes down as it moves from left to right, and with the y-intercept, one can start plotting the line at the point \( (0, \frac{8}{3}) \) on the y-axis. This form is extremely useful as it provides an immediate visual reference for the line's behavior on a graph.