Question

Make any line (1) with the points. Then write down equation (2) to create a system of equations that is consistent and dependent. (1) -3x+4y=8

Answer 1

The line -3x + 4y = 8 represents equation (1). To create a consistent and dependent system of equations, we'll use this line along with another equation derived from manipulating the given equation (1).

Step-by-step explanation:

The equation of the line given, -3x + 4y = 8, can be rearranged into slope-intercept form (y = mx + b) by isolating y. First, add 3x to both sides: 4y = 3x + 8. Then, divide both sides by 4 to obtain y = (3/4)x + 2. This equation represents a line with a slope of 3/4 and a y-intercept of 2.

For a system of equations to be consistent and dependent, the two equations must be multiples or representations of each other. In this case, the line -3x + 4y = 8 and its rewritten form y = (3/4)x + 2 both represent the same line in different forms. They are dependent equations as they convey the same relationship graphically, resulting in an infinite number of solutions when plotted on the coordinate plane. Any solution that satisfies one equation will automatically satisfy the other, leading to a consistent but dependent system.

Answer 2

To create a consistent and dependent system of equations with -3x + 4y = 8, multiply the entire equation by a non-zero constant. For example, multiplying by 2 gives -6x + 8y = 16, representing the same line and thus creating a system with infinitely many solutions.

Step-by-step explanation:

To answer the student's question, we need to construct a second linear equation that is both consistent and dependent on the provided equation -3x + 4y = 8. A consistent and dependent system of equations is one where both equations represent the same line, hence they have an infinite number of solutions. To create such an equation, we can multiply the given equation by any non-zero constant. Let's choose 2 for this example.

Multiplying the given equation by 2 yields: -6x + 8y = 16. This new equation represents the same line as the original equation and thus forms a consistent and dependent system of equations with it.

It is important to note that in order for two linear equations to form a dependent system, they must have the same ratio between their respective coefficients. That is, the ratios of the coefficients of x, the coefficients of y, and the constant terms must all be equal.