Question

1. if the equations of a system are 3x + 2y = 6 and 6x + 4y = 12, the system is:

a. consistent and independent
b. undefined
c. consistent and dependent
d. inconsistent

1. what does it mean if a system of equations has infinitely many solutions?

a. the equations do not intersect
b. the equations intersect at multiple points
c. the equations are perpendicular
d. the equations are the same line

1. what does the solution to a system of linear equations represent?

a. the point where the equations intersect
b. the slope of the line
c. the maximum value of the function
d. the y - intercept of the line

Explanation:

Question 7

Step1: Simplify the second equation

Divide the second equation \(6x + 4y=12\) by \(2\). We get \(3x + 2y = 6\), which is the same as the first equation.

Step2: Determine the system type

Since the two equations are essentially the same (one is a multiple of the other), the system has infinitely many solutions. A system with infinitely many solutions is consistent (because there are solutions) and dependent (because the equations are not distinct).

If a system of equations has infinitely many solutions, it means that the two equations represent the same line. Every point on that line is a solution. If equations do not intersect, the system is inconsistent. If they intersect at multiple points (for non - linear equations) or at one point (for linear equations), it's not the case of infinitely many solutions. Perpendicular lines intersect at exactly one point.

The solution to a system of linear equations is the set of values for the variables that satisfy all equations simultaneously. Geometrically, for two - variable linear equations (represented as lines), this is the point where the two lines intersect. The slope of a line is given by \(m=\frac{y_2 - y_1}{x_2 - x_1}\) for two points \((x_1,y_1)\) and \((x_2,y_2)\) on the line. The maximum value of a linear function (which is unbounded unless in a constrained optimization problem) is not relevant here. The \(y\) - intercept is the value of \(y\) when \(x = 0\) for a single line.

Answer:

c. Consistent and dependent

Question 8