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How do you solve $4 x + y = 4$ $4x + y = 4$ and $2 x + 8 y = 0$ $2x + 8y = 0$ using substitution?

2 Answers

Apr 9, 2016

$(x, y) = (\frac{16}{15}, - \frac{4}{15})$ $(x,y)=(16/15,-4/15)$
(see below for solution method using substitution)

Explanation:

Given: $4 x + y = 4$ $4x+y=4$

This implies: $y = 4 - 4 x$ $y=4-4x$
So we can substitute $(4 - 4 x)$ $(4-4x)$ for $y$ $y$
in the second given equation: $2 x + 8 y = 0$ $2x+8y=0$
to get:
$XXX 2 x + 8 (4 - 4 x) = 0$ $color(white)("XXX")2x+8(4-4x)=0$

Simplifying:
$XXX 2 x + 32 - 32 x = 0$ $color(white)("XXX")2x+32-32x=0$

$XXX 30 x = 32$ $color(white)("XXX")30x=32$

$XXX x = \frac{16}{15}$ $color(white)("XXX")x=16/15$

The we can substitute $\frac{16}{15}$ $16/15$ for $x$ $x$
in the first given equation: $4 x + y = 4$ $4x+y=4$
to get:
$XXX 4 \cdot (\frac{16}{15}) + y = 4$ $color(white)("XXX")4*(16/15)+y=4$

$XXX y = 4 - \frac{64}{15} = - \frac{4}{15}$ $color(white)("XXX")y=4-64/15 = -4/15$

ϕ

$phi$

Apr 9, 2016

$(x, y) = (\frac{16}{15}, - \frac{4}{15})$ $(x,y)=(16/15,-4/15)$

Explanation:

Solve by substitution and elimination

$4 x + y = 4$ $color(blue)(4x+y=4$

$2 x + 8 y = 0$ $color(blue)(2x+8y=0$

We can eliminate $4 x$ $4x$ from the first equation by $2 x$ $2x$ in the second equation (whole equation) with $- 2$ $-2$ to get $- 4 x$ $-4x$

$\to - 2 (2 x + 8 y = 0)$ $rarr-2(2x+8y=0)$

Use the distributive property

$a (b + c = x) = a b + a c = a x$ $color(brown)(a(b+c=x)=ab+ac=ax$

$\to - 4 x - 16 y = 0$ $rarr-4x-16y=0$

Now, add the above equation with the first equation to eliminate $4 x$ $4x$

$\to (4 x + y = 4) + (- 4 x - 16 y = 0)$ $rarr(4x+y=4)+(-4x-16y=0)$

$\to - 15 y = 4$ $rarr-15y=4$

Divide both sides by $- 15$ $-15$

$rarr(cancel(-15)y)/cancel(-15)=4/-15$

$color(green)(rArry=-4/15$

Now,substitute the value of $y$ to the first equation

$rarr4x+(-4/15)=4$

$rarr4x-4/15=4$

Add $4/15$ both sides

$rarr4x-4/15+4/15=4+4/15$

$rarr4x=64/15$

Divide both sides by $4$

$rarr(cancel4x)/cancel4=64/15-:4$

(I have written $64/15/4$ as $64/15-:4$ )

Take the reciprocal and multiply

$rarrx=cancel64^16/15*1/cancel4^1$

$color(green)(rArrx=16/15$

$color(blue)(ul bar |(x,y)=(16/15,-4/15)|$

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