True or False. The following relation represents a function.

What is the equation of the line that passes through (-2,3) and is parallel to 2x+3y=6?

Vsevolod [243]

Answer:

Step-by-step explanation:

$\bold{METHOD\ 1:}$

The slope-intercept form of an equation of a line:

$y=mx+b$

m - slope

b - y-intercept

Let $k:y=m_1x+b_1,\ l:y=m_2x+b_2$

then

$l\ \parallel\ k\iff m_1=m_2\\\\l\ \perp\ k\iff m_1m_2=-1\to m_2=-\dfrac{1}{m_1}$

We have the equation of a line:

$2x+3y=6$

Convert to the slope-intercept form:

$2x+3y=6$ subtract 2x from both sides

$3y=-2x+6$ divide both sides by 3

$y=-\dfrac{2}{3}x+2\to m_1=-\dfrac{2}{3}$

therefore the slope is $m_2=-\dfrac{2}{3}$

Put the value of the slope and the coordinates of the point (-2, 3) to the equation of a line:

$3=-\dfrac{2}{3}(-2)+b$

$3=\dfrac{4}{3}+b$ subtract 4/3 from both sides

$\dfrac{9}{3}-\dfrac{4}{3}=b\to b=\dfrac{5}{3}$

Finally:

$y=-\dfrac{2}{3}x+\dfrac{5}{3}$

Convert to the standard form (Ax + By = C):

$y=-\dfrac{2}{3}x+\dfrac{5}{3}$ multiply both sides by 3

$3y=-2x+5$ add 2x to both sides

$2x+3y=5$

$\bold{METHOD\ 2:}$

Let $k:A_1x+B_1y=C_1,\ l:A_2x+B_2y=C_2$ .

Lines k and l are parallel iff

$A_1=A_2\ \wedge\ B_1=B_2\to\dfrac{A_2}{A_1}=\dfrac{B_2}{B_1}$

We have the equation:

$2x+3y=6\to A_1=2,\ B_1=3$

then the equation of a line parallel to given lines has the equation:

$2x+3y=C$

Put the coordinates of the point (-2, 3) to the equation:

$C=2(-2)+3(3)\\\\C=-4+9\\\\C=5$

Finally:

$2x+3y=5$

4 0

3 years ago

Check if the following equality is true for all values of variables: (a–3c)(4c+2a)+3c(a+3c)=(2a–c)(3c+5a)–8a^2

BabaBlast [244]

Answer:

True for all values of a, b and c.

Step-by-step explanation:

(a–3c)(4c+2a)+3c(a+3c)=(2a–c)(3c+5a)–8a^2

Left side:

(a–3c)(4c+2a)+3c(a+3c)

= 4ac + 2a^2 - 12c^2 - 6ac + 3ac + 9c^2

= 2a^2 + ac - 3c^2

Right side:

(2a–c)(3c+5a)–8a^2

= 6ac + 10a^2 - 3c^2 - 5ac - 8a^2

= 2a^2 + ac - 3c^2.

So we see that the left side is identical to the right side so it is true for all values of the variables.

3 0

3 years ago

Simplify: 5x + 4 + 9x

Klio2033 [76]

The correct answer is; 14x +4

8 0

3 years ago

Read 2 more answers

If eight cars are entered in a race and three finishing places are considered, how many finishing orders can they finish? Assume

Sladkaya [172]

Answer:

a) 336

b)6

c)6

Step-by-step explanation:

a) In the 1st place fit 8 cars, in the 2nd only fit 7 cars, because one is already in the first place and 3th fit 6 cars, so all the possibilities are 8*7*6.

b) in the 1st place fit 3 cars, 2nd 2 cars and 3th 1 car, so all the possibilities are 3*2*1

c) every shirt can be matched with 1 pair of pants, there are 2 shirts and 3 pants, so all the the possibilities are 6+6.

3 0

4 years ago

HELP ASAP!!

GuDViN [60]

Step-by-step explanation:

Xm = (-3+3)/2 = 0/2 = 0

Ym = (2-2)/2 = 0/2 =0

so, the midpoint (Xm, Ym) is (0 , 0)

7 0

3 years ago

Read 2 more answers

True or False.The following relation represents a function. ​

True or False.The following relation represents a function.