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Math Expert V
Joined: 02 Sep 2009
Posts: 58421
If x + y = 4a/5, y + z = 7a/5 and z + x = 9a/5, then x + y + z =  [#permalink]

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Difficulty:   15% (low)

Question Stats: 85% (01:16) correct 15% (01:08) wrong based on 39 sessions

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If $$x + y = \frac{4a}{5}$$, $$y + z = \frac{7a}{5}$$ and $$z + x = \frac{9a}{5}$$, then $$x + y + z =$$

(A) 7a/15
(B) a
(C) 2a
(D) 3a
(E) 4a

Source: Nova GMAT
Difficulty Level: 550

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Originally posted by Bunuel on 01 Jun 2017, 12:51.
Last edited by SajjadAhmad on 07 Jul 2019, 01:22, edited 1 time in total.
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Re: If x + y = 4a/5, y + z = 7a/5 and z + x = 9a/5, then x + y + z =  [#permalink]

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1
Given that
$$x + y = \frac{4a}{5}$$, $$y + z = \frac{7a}{5}$$ and $$z + x = \frac{9a}{5}$$

Adding all 3, we get
$$2 * (x + y + z) = \frac{4a}{5} + \frac{7a}{5} + \frac{9a}{5}$$
$$2 * (x + y + z) = \frac{(4a+7a+9a)}{5}$$
$$x + y + z = \frac{20a}{5}*\frac{1}{2}$$
$$x + y + z = 2a$$ (Option C)
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Senior SC Moderator V
Joined: 22 May 2016
Posts: 3565
Re: If x + y = 4a/5, y + z = 7a/5 and z + x = 9a/5, then x + y + z =  [#permalink]

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Bunuel wrote:
If $$x + y = \frac{4a}{5}$$, $$y + z = \frac{7a}{5}$$ and $$z + x = \frac{9a}{5}$$, then $$x + y + z =$$

(A) 7a/15
(B) a
(C) 2a
(D) 3a
(E) 4a

pushpitkc 's method is quick; it works well in this three-equation/three variable problem where each variable has the same coefficient.

Another way to solve: use two different equations to isolate one variable. This method is handy when the variables' coefficients in the different equations are not "nice" numbers and typically need manipulating.

Given

$$x + y = \frac{4a}{5}$$

Rewrite: 5x + 5y = 4a --------------(P)

$$y + z = \frac{7a}{5}$$

Rewrite: 5y + 5z = 7a --------------(Q)

$$z + x = \frac{9a}{5}$$

Rewrite 5x + 5z = 9a ---------------(R)

So x + y + z = ?

1. To eliminate x, and to get another equation with y and z, subtract (P) from (R):

5x + 5z = 9a
-5x - 5y = -4a
5z - 5y = 5a ----------------------------(S)

2. Add new equation (S), and (Q) from above:

5z - 5y = 5a
5z + 5y = 7a
10z + 0y = 12a, hence

z= $$\frac{12}{10}$$a ------------------------------------> z = $$\frac{6}{5}$$a

3. Find y

From original (Q) y + z = $$\frac{7a}{5}$$

y = $$\frac{7a}{5}$$ - $$\frac{6a}{5}$$ =$$\frac{a}{5}$$-------------------------> y = $$\frac{a}{5}$$

4. Find x

From original (P) x + y =$$\frac{4a}{5}$$

x = $$\frac{4a}{5}$$ - $$\frac{a}{5}$$ = $$\frac{3a}{5}$$ ------------------------> x = $$\frac{3a}{5}$$

5. x + y + z = $$\frac{3a}{5}$$ + $$\frac{a}{5}$$ + $$\frac{6a}{5}$$

= $$\frac{10a}{5}$$, or 2a

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Joined: 26 Jul 2016
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Re: If x + y = 4a/5, y + z = 7a/5 and z + x = 9a/5, then x + y + z =  [#permalink]

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Adding all 3 gives 2(x+y+z)=20a/5----> x+y+z=2a
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Re: If x + y = 4a/5, y + z = 7a/5 and z + x = 9a/5, then x + y + z =  [#permalink]

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Bunuel wrote:
If $$x + y = \frac{4a}{5}$$, $$y + z = \frac{7a}{5}$$ and $$z + x = \frac{9a}{5}$$, then $$x + y + z =$$

(A) 7a/15
(B) a
(C) 2a
(D) 3a
(E) 4a

$$( x + y ) + ( y + z ) + ( z + x ) = \frac{4a}{5} + \frac{7a}{5} + \frac{9a}{5}$$

Or, $$2x + 2y + 2z = \frac{4a+ 7a + 9a}{5}$$

Or, $$2 ( x + y + z ) = \frac{20a}{5}$$

Or, $$2 ( x + y + z ) = 4a$$

Or, $$( x + y + z ) = 2a$$

Hence, answer must be (C) 2a
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Re: If x + y = 4a/5, y + z = 7a/5 and z + x = 9a/5, then x + y + z =  [#permalink]

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Bunuel wrote:
If $$x + y = \frac{4a}{5}$$, $$y + z = \frac{7a}{5}$$ and $$z + x = \frac{9a}{5}$$, then $$x + y + z =$$

(A) 7a/15
(B) a
(C) 2a
(D) 3a
(E) 4a

Adding our 3 equations together, we get:

2x + 2y + 2z = 4a/5 + 7a/5 + 9a/5

2x + 2y + 2z = (4a + 7a + 9a)/5

2x + 2y + 2z = (20a)/5 = 4a

Dividing the entire equation by 2, we have:

x + y + z = 2a

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Re: If x + y = 4a/5, y + z = 7a/5 and z + x = 9a/5, then x + y + z =  [#permalink]

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_________________ Re: If x + y = 4a/5, y + z = 7a/5 and z + x = 9a/5, then x + y + z =   [#permalink] 24 Feb 2019, 13:40
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