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# System

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x+x+y=14
L+L+z=20
x+y+z=14
x+z+L=18
If numbers x,y,z,L satisfy the equations above, find the value of x+z.

May 31, 2022

#1
+1

Solve the following system:
{y + 2 x = 14 | (equation 1)
z + 2 L = 20 | (equation 2)
x + y + z = 14 | (equation 3)
L + x + z = 18 | (equation 4)

Subtract 1/2 × (equation 1) from equation 3:
{2 x + y + 0 z + 0 L = 14 | (equation 1)
0 x + 0 y + z + 2 L = 20 | (equation 2)
0 x + y/2 + z + 0 L = 7 | (equation 3)
x + 0 y + z + L = 18 | (equation 4)

Multiply equation 3 by 2:
{2 x + y + 0 z + 0 L = 14 | (equation 1)
0 x + 0 y + z + 2 L = 20 | (equation 2)
0 x + y + 2 z + 0 L = 14 | (equation 3)
x + 0 y + z + L = 18 | (equation 4)

Subtract 1/2 × (equation 1) from equation 4:
{2 x + y + 0 z + 0 L = 14 | (equation 1)
0 x + 0 y + z + 2 L = 20 | (equation 2)
0 x + y + 2 z + 0 L = 14 | (equation 3)
0 x - y/2 + z + L = 11 | (equation 4)

Multiply equation 4 by 2:
{2 x + y + 0 z + 0 L = 14 | (equation 1)
0 x + 0 y + z + 2 L = 20 | (equation 2)
0 x + y + 2 z + 0 L = 14 | (equation 3)
0 x - y + 2 z + 2 L = 22 | (equation 4)

Swap equation 2 with equation 3:
{2 x + y + 0 z + 0 L = 14 | (equation 1)
0 x + y + 2 z + 0 L = 14 | (equation 2)
0 x + 0 y + z + 2 L = 20 | (equation 3)
0 x - y + 2 z + 2 L = 22 | (equation 4)

Add equation 2 to equation 4:
{2 x + y + 0 z + 0 L = 14 | (equation 1)
0 x + y + 2 z + 0 L = 14 | (equation 2)
0 x + 0 y + z + 2 L = 20 | (equation 3)
0 x + 0 y + 4 z + 2 L = 36 | (equation 4)

Divide equation 4 by 2:
{2 x + y + 0 z + 0 L = 14 | (equation 1)
0 x + y + 2 z + 0 L = 14 | (equation 2)
0 x + 0 y + z + 2 L = 20 | (equation 3)
0 x + 0 y + 2 z + L = 18 | (equation 4)

Swap equation 3 with equation 4:
{2 x + y + 0 z + 0 L = 14 | (equation 1)
0 x + y + 2 z + 0 L = 14 | (equation 2)
0 x + 0 y + 2 z + L = 18 | (equation 3)
0 x + 0 y + z + 2 L = 20 | (equation 4)

Subtract 1/2 × (equation 3) from equation 4:
{2 x + y + 0 z + 0 L = 14 | (equation 1)
0 x + y + 2 z + 0 L = 14 | (equation 2)
0 x + 0 y + 2 z + L = 18 | (equation 3)
0 x + 0 y + 0 z + (3 L)/2 = 11 | (equation 4)

Multiply equation 4 by 2:
{2 x + y + 0 z + 0 L = 14 | (equation 1)
0 x + y + 2 z + 0 L = 14 | (equation 2)
0 x + 0 y + 2 z + L = 18 | (equation 3)
0 x + 0 y + 0 z + 3 L = 22 | (equation 4)

Divide equation 4 by 3:
{2 x + y + 0 z + 0 L = 14 | (equation 1)
0 x + y + 2 z + 0 L = 14 | (equation 2)
0 x + 0 y + 2 z + L = 18 | (equation 3)
0 x + 0 y + 0 z + L = 22/3 | (equation 4)

Subtract equation 4 from equation 3:
{2 x + y + 0 z + 0 L = 14 | (equation 1)
0 x + y + 2 z + 0 L = 14 | (equation 2)
0 x + 0 y + 2 z + 0 L = 32/3 | (equation 3)
0 x + 0 y + 0 z + L = 22/3 | (equation 4)

Divide equation 3 by 2:
{2 x + y + 0 z + 0 L = 14 | (equation 1)
0 x + y + 2 z + 0 L = 14 | (equation 2)
0 x + 0 y + z + 0 L = 16/3 | (equation 3)
0 x + 0 y + 0 z + L = 22/3 | (equation 4)

Subtract 2 × (equation 3) from equation 2:
{2 x + y + 0 z + 0 L = 14 | (equation 1)
0 x + y + 0 z + 0 L = 10/3 | (equation 2)
0 x + 0 y + z + 0 L = 16/3 | (equation 3)
0 x + 0 y + 0 z + L = 22/3 | (equation 4)

Subtract equation 2 from equation 1:
{2 x + 0 y + 0 z + 0 L = 32/3 | (equation 1)
0 x + y + 0 z + 0 L = 10/3 | (equation 2)
0 x + 0 y + z + 0 L = 16/3 | (equation 3)
0 x + 0 y + 0 z + L = 22/3 | (equation 4)

Divide equation 1 by 2:
{x + 0 y + 0 z + 0 L = 16/3 | (equation 1)
0 x + y + 0 z + 0 L = 10/3 | (equation 2)
0 x + 0 y + z + 0 L = 16/3 | (equation 3)
0 x + 0 y + 0 z + L = 22/3 | (equation 4)

Collect results:

x = 16/3
y = 10/3
z = 16/3
L = 22/3                            x  +  z ==16/3  +  16/3==32/3

May 31, 2022
#2
+2448
0

We have the system of equations:

$$2x + y = 14$$

$$2L + z = 20$$

$$x +y + z = 14$$

$$x + z + L = 18$$

Subtracting $$y$$ from both the first and third equations gives us: $$x + z = 14 - y$$ and $$2x = 14-y$$

Because both equations are equal to each other, we can set them equal. This gives us: $$x + z = 2x$$, meaning $$x = z$$

Substituting this in, we have a new system in terms of $$x$$:

$$2x + y = 14$$

$$2L + x = 20$$

$$2x + L = 18$$

Now, we are looking to find the value of $$2x$$

We can do this by disregarding the first equation, which gives us this system:

$$2L + x = 20$$

$$2x + L = 18$$

Multiplying the second equation by 2 gives us: $$4x + 2L = 36$$

Now, subtracting the first equation from this shows: $$3x = 16$$.

Multiplying both sides by $$2 \over 3$$ shows $$x +z = \color{brown}\boxed{32 \over 3}$$, just as Guest found

May 31, 2022
edited by BuilderBoi  May 31, 2022