Questions and Answers #2 Linear Equations and Graphs

Questions and Answers Sheet 2 Linear Equations and Graphs Question #1 Which of the subsets of 𝑅3×3 are subspaces of 𝑅3×3 ? a) The invertible 3 × 3 matrices b) The 3 × 3 matrices whose entries are all integers c) The 3 × 3 matrices with all zeros in the third row d) The non-invertible 3 × 3 matrices e) The diagonal 3 × 3 matrices f) The symmetric 3 × 3 matrices Answer: a) No, 'cause it is not invertible b) No, 'cause they are not closed under scalar multiplication over R c) I would say yes... check it by hands! d) No, the sum of two of them may be invertible e) Yes!! Easy to check f) I would say ''YES'', but again, try to check it by hands! Question #2 Tell, what is the slope and y-intercept of this line 𝑦 = −2 ? Answer: 𝑦 = −2 is a line crossing the point at (0, −2) . Thus, the gradient is zero and the y intercept is -2. Question #3 How to graph the line 𝑦 = −2 Answer: It doesn't matter what value you put in for x, the y is always going to be -2. So, first plot (0, −2) Then, you plot any point you'd like, as long as the y value is -2. 𝑦 = −2 is a straight horizontal line that crosses the y axis at -2. Question #4 How do you graph 𝑥 ≻ 2 on a coordinate plane? Answer: 𝑥 = −2 is a graph that is a vertical line with all x-coordinates equal to -2. Question #5 How to graph 𝑥 = −3 ? Answer: It's the equation of a vertical line passing through 𝑥 = −3 Question #6 How can I graph 𝑦 = 2𝑥 − 1 by plotting points? Answer: To find the y-intercept, put 𝑥 = 0 𝑦 = 2(0) − 1 𝑦 = −1 To find the x-intercept, put 𝑦 = 0 2𝑥 − 1 = 0 2𝑥 = 1 1 𝑥= 2 Draw a system of axis. Mark these points on the graph. Now draw a line through these two ponts. To draw the graph by used points, let 𝑥 = −1 , then 𝑦 = 2(−1) − 1 = −2 − 1 = −3 Thus, first point is (−1; −3) Let 𝑥 = 0, 𝑦 = 2(0) − 1 = −1 And second point is (0; −1) Let 𝑥 = 1, 𝑦 = 2(1) − 1 = 1 Third point is (1; 1) Here you go. Question #7 Explain how to graph the line 𝑥 − 𝑦 = 2 Answer: We have 𝑥 − 𝑦 = 2 𝑥−𝑦+𝑦 =2+𝑦 𝑥+0=2+𝑦 𝑥−2=2−2+𝑦 𝑥−2=0+𝑦 𝑦 =𝑥−2 Now, select values for x at random to build the graph Let 𝑥 = −1 , then 𝑦 = −1 − 2 = −3 Let 𝑥 = 0 , then 𝑦 = 0 − 2 = −2 Let 𝑥 = 2 , then 𝑦 = 2 − 2 = 0 So, we have the points: (−1, −3), (0, −2), (2,0) Now mark these point on the graph and draw a line through them. Here you are. Question #8 Write the equation of the line with an undefined slope and goes through point (2,4) Answer: If the slope of the line is undefined, by definition the line is a vertical line. For a vertical line, the value of x is the same for each and every value of y. The equation of the line is: 𝑥 = 2 Question #9 How to graph 𝑦 = − 2 3 Answer: 𝑦=− 2 3 is an equation, which represents a horisontal line with a slope of zero parallel to the x-axis. 2 We have the garph 𝑦 − 0.001𝑥 + = 0[−10,10, −5,5] 3 Question #10 How to graph 6𝑥 + 5𝑦 = 20 ? Answer: First, find the equation's two solution points and plot them on the graph. First Point: For 𝑥 = 0 (6 ∗ 0) + 5𝑦 = 20 0 + 5𝑦 = 20 5𝑦 = 20 5 y 20 = 5 5 𝑦 = 4 or (0,4) Second Point: For 𝑦 = −2 6𝑥 + (5 ∗ −2) = 20 6𝑥 − 10 = 20 6x −10 + 10 = 20 + 10 6𝑥 − 0 = 30 6𝑥 = 30 6 x 30 = 6 6 𝑥 = 5 or (5, −2) We can next plot the two points on the coordinate plane: graph(𝑥 2 + (𝑦 − 4)2 − 0.075)((𝑥 − 5)2 + (𝑦 + 2)2 − 0.075) = 0[−20,20, −10,10] Now, we can draw a straight line through the two points to graph the line: graph(6𝑥 + 5𝑦 − 20)(𝑥 2 + (𝑦 − 4)2 − 0.075)((𝑥 − 5)2 + (𝑦 + 2)2 − 0.075) = 0[−20,20, −10,10]