Expert Answers

1.True or false: A system comprising two linear equations in two variables has a unique solution if and only if the straight lines represented by the equations are nonparallel.

2.True or false: Suppose the straight lines represented by a system of two linear equations in two variables are parallel to each other. Then the system has infinitely many solutions.

3.True or false: If A and B are matrices of the same order, then (A+B)T=AT+BT.

4.True or false: If A and B are matrices such that AB and BA are both defined, then A and B must be square matrices.

5.True or false: If A is a square matrix with inverse ( A^-1) and c is a nonzero real number, then

6.True or false: If AX = B is a system of n linear equations in n unknowns and ( A^-1) does not exist, then AX = B does not have a unique solution.

If you think the statement is true, then prove it. On the other hand, if you think the statement is false, then give an example that disproves the statement. For example, the statement “If a and b are real numbers, then a – b = b – a” is false and an example that disproves it may be constructed by taking a = 3 and b = 5. For these values of a and b, we find a – b = 3 – 5 = -2 but b – a = 5 – 3 = 2 and this shows that a – b ≠ b – a. Such an example is called a ≠counterexample.

1.True or false. The slope of a horizontal line is undefined.
2.True or false. Suppose the slope of a straight line L is -3/4 and P is a given point on L. If Q is a point on L lying 2 units to the right of P, then Q is situated 3/2 units below P.
3.True or false. The y-intercept of the straight line with equation Ax + By + C = 0 is -C/B (B ≠ 0).
4.True or false. If a line L1 has equation y = mx + b, where m and b are constants with m ≠ 0,then an equation of a line L2 perpendicular to L1 has the form , where C is a constant.
5.True or false. Suppose an asset is being depreciated linearly. Then the rate of depreciation of the asset is given by the negative of the slope of the depreciation line.
6.True or false. If R and C are linear revenue and cost functions, respectively, and (x0, p0) is the breakeven point, then P(x) > P(x0) if x > x0, where P is the profit function.
7.True or false. The least-squares line must pass through at least one of the data points.