# Determine whether the given (2×3)(2×3) system of linear equations represents coincident planes (that is, the same plane), two parallel planes, or two planes whose intersection is a line. In the latter case, give the parametric equations for the line, that is, give equations of the form x=at+b , y=ct+d , z=et+f 2x_1+x_2+x_3=3 -2x_1+x_2-x_3=1

Question
Forms of linear equations
Determine whether the given (2×3)(2×3) system of linear equations represents coincident planes (that is, the same plane), two parallel planes, or two planes whose intersection is a line. In the latter case, give the parametric equations for the line, that is, give equations of the form
$$x=at+b , y=ct+d , z=et+f$$
$$2x_1+x_2+x_3=3$$
$$-2x_1+x_2-x_3=1$$

2021-02-10
The given system of linear equations
$$2x_1+x_2+x_3=3$$
$$-2x_1+x_2+x_3=1$$
We need to determine whether the given system of linear equations represents coincident planes, two parallel planes, or two planes whose intersection is a line.
Let $$n_1 = (2,1,1)$$ and $$n_2 = (-2,1,-1)$$ be normal vectors of both the equations. Then n, and na are not parallel.
Let $$x_3 = t$$, then replace $$x_3 = t$$ in the system of linear equations $$2x_1+x_2+t=3 \ \ \ (1)$$
$$-2x_1+x_2+t=1 \ \ \ (2)$$
$$(1)+(2) \Rightarrow 2x_2+2t=4 \Rightarrow 2x_2=4-2t \Rightarrow x_2=2-t$$ Replace $$x_2 = 2-t$$ and $$x_3 =t$$ in the firs equation
$$2x_1+2-t+t=3$$
$$2x_1=3-2$$
$$x_1=\frac{1}{2}$$
Hence, the plane intersect in a line and the parametric equations are
$$x_1=\frac{1}{2} , x_2=2-t \text{ and } x_3=t$$

### Relevant Questions

Determine whether the given $$\displaystyle{\left({2}\ \times\ {3}\right)}$$ system of linear equations represents coincident planes (that is, the same plane), two parallel planes, or two planes whose intersection is a line. In the latter case, give the parametric equations for the line, that is, give equations of the form
$$\displaystyle{x}={a}{t}\ +\ {b},\ {y}={c}{t}\ +\ {d},\ {z}={e}{t}\ +\ {f}.$$
$$\displaystyle{x}_{{{1}}}\ +\ {2}{x}_{{{2}}}\ -\ {x}_{{{3}}}={2}$$
$$\displaystyle{x}_{{{1}}}\ +\ {x}_{{{2}}}\ +\ {x}_{{{3}}}={3}$$

A line L through the origin in $$\displaystyle\mathbb{R}^{{3}}$$ can be represented by parametric equations of the form x = at, y = bt, and z = ct. Use these equations to show that L is a subspase of $$RR^3$$  by showing that if $$v_1=(x_1,y_1,z_1)\ and\ v_2=(x_2,y_2,z_2)$$  are points on L and k is any real number, then $$kv_1\ and\ v_1+v_2$$  are also points on L.

(a) Find a system of two linear equations in the variables x and y whose solution set is given by the parametric equations $$x = t$$ and y $$= 3- 2t.$$
(b) Find another parametric solution to the system in part (a) in which the parameter is s and $$y =s.$$

A random sample of $$n_1 = 14$$ winter days in Denver gave a sample mean pollution index $$x_1 = 43$$.
Previous studies show that $$\sigma_1 = 19$$.
For Englewood (a suburb of Denver), a random sample of $$n_2 = 12$$ winter days gave a sample mean pollution index of $$x_2 = 37$$.
Previous studies show that $$\sigma_2 = 13$$.
Assume the pollution index is normally distributed in both Englewood and Denver.
(a) State the null and alternate hypotheses.
$$H_0:\mu_1=\mu_2.\mu_1>\mu_2$$
$$H_0:\mu_1<\mu_2.\mu_1=\mu_2$$
$$H_0:\mu_1=\mu_2.\mu_1<\mu_2$$
$$H_0:\mu_1=\mu_2.\mu_1\neq\mu_2$$
(b) What sampling distribution will you use? What assumptions are you making? NKS The Student's t. We assume that both population distributions are approximately normal with known standard deviations.
The standard normal. We assume that both population distributions are approximately normal with unknown standard deviations.
The standard normal. We assume that both population distributions are approximately normal with known standard deviations.
The Student's t. We assume that both population distributions are approximately normal with unknown standard deviations.
(c) What is the value of the sample test statistic? Compute the corresponding z or t value as appropriate.
(Test the difference $$\mu_1 - \mu_2$$. Round your answer to two decimal places.) NKS (d) Find (or estimate) the P-value. (Round your answer to four decimal places.)
(e) Based on your answers in parts (i)−(iii), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level \alpha?
At the $$\alpha = 0.01$$ level, we fail to reject the null hypothesis and conclude the data are not statistically significant.
At the $$\alpha = 0.01$$ level, we reject the null hypothesis and conclude the data are statistically significant.
At the $$\alpha = 0.01$$ level, we fail to reject the null hypothesis and conclude the data are statistically significant.
At the $$\alpha = 0.01$$ level, we reject the null hypothesis and conclude the data are not statistically significant.
(f) Interpret your conclusion in the context of the application.
Reject the null hypothesis, there is insufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Reject the null hypothesis, there is sufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Fail to reject the null hypothesis, there is insufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Fail to reject the null hypothesis, there is sufficient evidence that there is a difference in mean pollution index for Englewood and Denver. (g) Find a 99% confidence interval for
$$\mu_1 - \mu_2$$.
lower limit
upper limit
(h) Explain the meaning of the confidence interval in the context of the problem.
Because the interval contains only positive numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is greater than that of Denver.
Because the interval contains both positive and negative numbers, this indicates that at the 99% confidence level, we can not say that the mean population pollution index for Englewood is different than that of Denver.
Because the interval contains both positive and negative numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is greater than that of Denver.
Because the interval contains only negative numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is less than that of Denver.
What are the parametric equations for the intersection of the planes x-y-z=1 and 2x + 3y + z = 2 ?
Consider the curves in the first quadrant that have equationsy=Aexp(7x), where A is a positive constant. Different valuesof A give different curves. The curves form a family,F. Let P=(6,6). Let C be the number of the family Fthat goes through P.
A. Let y=f(x) be the equation of C. Find f(x).
B. Find the slope at P of the tangent to C.
C. A curve D is a perpendicular to C at P. What is the slope of thetangent to D at the point P?
D. Give a formula g(y) for the slope at (x,y) of the member of Fthat goes through (x,y). The formula should not involve A orx.
E. A curve which at each of its points is perpendicular to themember of the family F that goes through that point is called anorthogonal trajectory of F. Each orthogonal trajectory to Fsatisfies the differential equation dy/dx = -1/g(y), where g(y) isthe answer to part D.
Find a function of h(y) such that x=h(y) is the equation of theorthogonal trajectory to F that passes through the point P.

1. S1 and S2, shown above, are thin parallel slits in an opaqueplate. A plane wave of wavelength λ is incident from the leftmoving in a direction perpendicular to the plate. On a screenfar from the slits there are maximums and minimums in intensity atvarious angles measured from the center line. As the angle isincreased from zero, the first minimum occurs at 3 degrees. Thenext minimum occurs at an angle of-
A. 4.5 degrees
B. 6 degrees
C. 7.5 degrees
D. 9 degrees
E. 12 degrees
For the equation (-1,2), $$y= \frac{1}{2}x - 3$$, write an equation in slope intercept form for the line that passes through the given point and is parallel to the graph of the given equation.
Write the homogeneous system of linear equations in the form AX = 0. Then verify by matrix multiplication that the given matrix X is a solution of the system for any real number $$c_1$$
$$\begin{cases}x_1+x_2+x_3+x_4=0\\-x_1+x_2-x_3+x_4=0\\ x_1+x_2-x_3-x_4=0\\3x_1+x_2+x_3-x_4=0 \end{cases}$$
$$X =\begin{pmatrix}1\\-1\\-1\\1\end{pmatrix}$$
Let B be a $$(4\times3)(4\times3)$$ matrix in reduced echelon form.
b) Suppose that a system of 4 linear equations in 2 unknowns has augmented matrix A, where A is a $$(4\times3)(4\times3)$$ matrix row equivalent to B.