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How do you solve the system $4 x - 5 y = - 7$ $4x-5y=-7$ and $y = 5 x$ $y=5x$ using substitution?

1 Answer

Oct 1, 2016

$x = \frac{1}{3} and y = \frac{5}{3}$ $x = 1/3 and y = 5/3$

Explanation:

There are two variables, therefore we need two equations.

$4 x - 5 y = - 7$ $4x-5y=-7$ and $y = 5 x$ $color(red)(y=5x)$

$y = 5 x$ $color(red)(y=5x)$ means that $y and 5 x$ $y and 5x$ have exactly the same value.

$y and 5 x$ $color(red)(y" and " 5x)$ are therefore interchangeable.

$4 x - 5 y = - 7$ $4x-5color(red)(y)=-7$ has two variables and therefore cannot be solved with one unique solution. There are infinitely many answers.

Replace $y$ $color(red)(y)$ by $5 x$ $color(red)(5x)$
This gives an equation with only ONE variable and it can be solved.

$4 x - 5 y = - 7$ $" "4x-5color(red)(y)=-7$
$\times \times \times x ↓$ $color(white)(xxxxxxx)darr$
$\to 4 x - 5 (5 x) = - 7$ $rarr4x-5color(red)((5x))=-7" "$ (now the equation has only x)

$4 x - 25 x = - 7$ $4x -25x = -7$

$- 21 x = - 7 \leftarrow \div - 21$ $-21x = -7" "larr div -21$

$x = \frac{1}{3}$ $x = 1/3$

Now find the value of $y$ $y$ , knowing that $y = 5 x$ $y = 5x$

$y = 5 (\frac{1}{3})$ $y = 5(1/3)$

$y = \frac{5}{3}$ $y = 5/3$

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