# How do you solve x+2y=5 and 2x-3y=-4?

Jan 21, 2016

If P is the point where the two graphs cross then;

color(purple)(=> P_("crossing point") = (x,y)->(0.7,1.8) ->(7/10,9/5)

#### Explanation:

Given:
$\textcolor{b r o w n}{x + 2 y = 5}$................................(1)
$\textcolor{b r o w n}{2 x - 3 y = - 4}$........................(2)

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Using short cut methods: (jumping steps)

$\textcolor{b l u e}{\text{Making y the dependant variable:}}$
$y = - x + \frac{5}{2.} \ldots \ldots \ldots \ldots \ldots \ldots . \left({1}_{a}\right)$
$y = \frac{2}{3} x + \frac{4}{3.} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \left({2}_{a}\right)$

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$\textcolor{b l u e}{\text{To determine the value of } x}$

By substitution we can equate equation $\left({1}_{a}\right)$ to equation $\left({2}_{a}\right)$ through $y$ giving:

$- x + \frac{5}{2} = y = \frac{2}{3} x + \frac{4}{3}$

Collecting like terms

$\frac{2}{3} x + x = \frac{5}{2} - \frac{4}{3}$

$\frac{5}{3} x = \frac{15 - 8}{6}$

$\textcolor{b l u e}{x = \frac{7}{6} \times \frac{3}{5} = \frac{7}{10}}$...........................(3)
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$\textcolor{b l u e}{\text{To determine the value of } y}$

Substitute equation (3) into equation $\left({1}_{a}\right)$. The easier of the two!

$y = - \frac{7}{10} + \frac{5}{2}$

$y = \frac{25 - 7}{10}$

$y = \frac{18}{10} = \frac{9}{5}$

$\textcolor{b l u e}{y = \frac{9}{5}}$

Note: $y = \frac{9}{5} = 1.8 \text{ and } x = \frac{7}{10} = 0.7$

$\implies {P}_{\text{crossing point}} = \left(x , y\right) \to \left(0.7 , 1.8\right)$