When solving a system of equations using substitution, there are three steps.

Isolate a variable in one of the equations.
Substitute the expression for that variable into the other equation and solve.
Substitute this solution into one of the equations and solve for the value of the other variable

Consider the given equations, we need to isolate one of the variables. Let's start by isolating

x

in Equation I.

{2 = 2 y - x 23 = 5 y - 4 x (I) (II)

$(I):$ $LHS + x = RHS + x$

{x + 2 = 2 y 23 = 5 y - 4 x

$(I):$ $LHS - 2 = RHS - 2$

{x = 2 y - 2 23 = 5 y - 4 x

Now that we've isolated

x,

we can solve the system by substitution.

{x = 2 y - 2 23 = 5 y - 4 x (I) (II)

$(II):$ $x = 2 y - 2$

{x = 2 y - 2 23 = 5 y - 4 (2 y - 2)

$(II):$ Distribute $- 4$

{x = 2 y - 2 23 = 5 y - 8 y + 8

$(II):$ Subtract term

{x = 2 y - 2 23 = - 3 y + 8

$(II):$ $LHS - 8 = RHS - 8$

{x = 2 y - 2 15 = - 3 y

$(II):$ $LHS / (- 3) = RHS / (- 3)$

{x = 2 y - 2 - 5 = y

$(II):$ Rearrange equation

{x = 2 y - 2 y = - 5

Now, to find the value of

x,

we need to substitute

y = - 5

into either one of the equations in the given system. Let's use the first equation.

{x = 2 y - 2 y = - 5 (I) (II)

$(I):$ $y = - 5$

{x = 2 (- 5) - 2 y = - 5

$(I):$ $a (- b) = - a \cdot b$

{x = - 10 - 2 y = - 5

$(I):$ Subtract term

{x = - 12 y = - 5

The solution, or point of intersection, to this system of equations is the point

(- 12, - 5) .

To check our answer, we will substitute our solution into both equations. If doing so results in true statements, then our solution is correct.

{2 = 2 y - x 23 = 5 y - 4 x (I) (II)

$(I), (II):$ $x = - 12$ , $y = - 5$

{2 = ? 2 (- 5) - (- 12) 23 = ? 5 (- 5) - 4 (- 12)

$(I), (II):$ Multiply

{2 = ? - 10 + 12 23 = ? - 25 + 48

$(I), (II):$ Add terms

{2 = 2 23 = 23

Because both equations are true statements, we know that our solution is correct.

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