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2 years ago

Solve the system by substitution -x - y - z = -8 -4x + 4y + 5z = 7 2x + 2z = 4

Mathematics

2 answers:

zhuklara [117]2 years ago

7 0

Answer: The required solution of the given system is

x = 3, y = 6 and z = -1.

Step-by-step explanation: We are given to solve the following system of equations by the method of substitution :

$-x-y-z=-8~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)\\\\-4x+4y+5z=7~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(ii)\\\\2x+2z=4~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(iii)$

From equation (iii), we have

$2x+2z=4\\\\\Rightarrow x+z=2\\\\\Rightarrow x=2-z~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(iv)$

Substituting the value of x from equation (iv) in equations (i) and (ii), we get

$-(2-z)-y-z=-8\\\\\Rightarrow 2-z+y+z=8\\\\\Rightarrow 2+y=8\\\\\Rightarrow y=8-2\\\\\Rightarrow y=6$

and

$-4(2-z)+4y+5z=7\\\\\Rightarrow -8+4z+4\times6+5z=7\\\\\Rightarrow 9z+16=7\\\\\Rightarrow 9z=-9\\\\\Rightarrow z=-\dfrac{9}{9}\\\\\Rightarrow z=-1.$

From equation (iii), we get

$2x+2(-1)=4\\\\\Rightarrow 2x=6\\\\\Rightarrow x=\dfrac{6}{2}\\\\\Rightarrow x=3.$

Thus, the required solution of the given system is

x = 3, y = 6 and z = -1.

gulaghasi [49]2 years ago

5 0

Answer: 14

Step-by-step explanation:

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The area of the triangle shown is 40.0 cm2.

kompoz [17]

Answer:

The height of the triangle is 6.4cm

Step-by-step explanation:

- If the triangle has a side that measures 12.5cm, it means that two of its sides are equal in length.

Which is equivalent to the description of an isosceles triangle.

To find the height, use the equation of the area.

Area = (Base * height) / 2

- We know the value of the area and its shorter side that could be the base.

40cm = (12.5 cm * h) / 2

- Clearing the height would give that:

80cm / 12.5cm = h

h = 6.4cm

The height of the triangle is 12.5cm

4 0

1 year ago

Pedro has created the function f(x)= 4x-3/2 to represent the number of assingments he has completed where x represents the numbe

Law Incorporation [45]

The given function is
f(x) = 4x - 3/2
where
f(x) = number of assignments completed
x = number of weeks required to complete the assignments

We want to find f⁻¹ (30) as an estimate of the number of weeks required to complete 30 assignments.
The procedure is as follows:

1. Set y = f(x)
y = 4x - 3/2

2. Exchange x and y
x = 4y - 3/2

3. Solve for y
4y = x + 3/2
y = (x +3/2)/4

4. Set y equal to f⁻¹ (x)
f⁻¹ (x) = (x + 3/2)/4

5. Find f⁻¹ (30)
f⁻¹ (30) = (30 + 3/2)/4 = 63/8 = 8 (approxmately)

Answer:
Pedro needs about 8 weeks to complete 30 assignments.

6 0

2 years ago

12. A cyclist travels 40 km at a speed of x km/h. Find the time

GREYUIT [131]

Answer:

Step-by-step explanation:

distance = rate times time, so

distance

time = --------------

rate

Here the distance is 40 km and the rate is x. Then

40 km

time = -------------- = (40/x) hr

x km/hr

If the speed is reduced by 2 km/hr, we get:

40 km 40 km 40

time = ------------------------------- or time = ----------------------- = --------- hr

(x km/hr - 2 km/hr) (x - 2) (km/hr) x - 2

The difference between the times is:

40 40

-------- - --------- = 1 hr Solve this for the original speed, x

x - 2 x

40x - 40x + 80 80

----------------------- = 1 hr or ------------ = 1 or 80 = x^2 - 2x

x(x - 2) x(x - 2)

Rewriting this in standard quadratic form: x^2 - 2x - 80 = 0

Here a = 1, b = -2 and c = -80, and so the discriminant b^2 - 4ac is:

(-2)^2 - 4(1)(-80) = 324

The solutions are:

-(-2) ± √324 2 ± 18

x = -------------------- = ------------ = 1 ± 9, or 10 (discard the other root)

2 2

The original speed was 10 km/hr

7 0

1 year ago

Read 2 more answers

Xavier is working two summer jobs, making $14 per hour lifeguarding and $15 per hour tutoring. Xavier must earn at least $170 th

Julli [10]

Answer:

Step-by-step explanation:

8 0

2 years ago

A random sample of 35 undergraduate students who completed two years of college were asked to take a basic mathematics test. The

lisov135 [29]

Answer:

$(75.1 -72.1) -1.66 \sqrt{\frac{12.8^2}{35} +\frac{14.6^2}{50}}= -1.96$

$(75.1 -72.1) +1.66 \sqrt{\frac{12.8^2}{35} +\frac{14.6^2}{50}}= 7.96$

And the confidence interval would be given by:

$-1.96 \leq \mu_1 -\mu_2 \leq 7.96$

And since the confidence interval contains the value 0 we have enough evidence to conclude that we don't have significant differences between the two means at 10% of significance.

Step-by-step explanation:

For this case we have the following info given :

$\bar X_1= 75.1$ represent the sample mean for the scores of the undergraduate students

$s_1 = 12.8$ represent the standard deviation for the undergraduate students

$n_1 =35$ the sample size for the undergraduate

$\bar X_2= 72.1$ represent the sample mean for the scores of the high school students

$s_2 = 14.6$ represent the standard deviation for the high school students

$n_2 =50$ the sample size for the high school

The confidence interval for the true difference of means is given by:

$(\bar X_1 -\bar X_2) \pm t_{\alpha/2} \sqrt{\frac{s^2_1}{n_1} +\frac{s^2_2}{n_2}}$

The degrees of freedom are given by:

$df=n_1 +n_2 -2= 35+50-2=83$

The confidence level is 90% and the significance level is $\alpha=0.1$ and $\alpha/2 =0.05$ then the critical value would be:

$t_{\alpha/2}= 1.99$

And replacing the info we got:

$(75.1 -72.1) -1.66 \sqrt{\frac{12.8^2}{35} +\frac{14.6^2}{50}}= -1.96$

$(75.1 -72.1) +1.66 \sqrt{\frac{12.8^2}{35} +\frac{14.6^2}{50}}= 7.96$

And the confidence interval would be given by:

$-1.96 \leq \mu_1 -\mu_2 \leq 7.96$

And since the confidence interval contains the value 0 we have enough evidence to conclude that we don't have significant differences between the two means at 10% of significance.

8 0

2 years ago