We want to find the values of $x$ and $y$ for which $R S T V$ is a parallelogram. To do so, we will use the given algebraic expressions for the segment lengths.

The Parallelogram Opposite Sides Theorem states that if a quadrilateral is a parallelogram then its opposite sides are congruent.

\overline{R V} ≅ \overline{S T} and \overline{V T} ≅ \overline{R S}

By the definition of congruent segments, we can conclude that their lengths are equal.

R V = S T and V T = R S

Let's create a system of equations by substituting the lengths of the segments into these equations.

{2 x + 3 = y + 5 5 x = 4 y - 1

To solve it we will use the Substitution Method.

{2 x + 3 = y + 5 5 x = 4 y - 1 (I) (II)

SubEqn

$(I):$ $LHS - 5 = RHS - 5$

{2 x - 2 = y 5 x = 4 y - 1

Substitute

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

{2 x - 2 = y 5 x = 4 (2 x - 2) - 1

▼

(II):

Solve for

x

Distr

$(II):$ Distribute $4$

{2 x - 2 = y 5 x = 8 x - 8 - 1

SubTerm

$(II):$ Subtract term

{2 x - 2 = y 5 x = 8 x - 9

SubEqn

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

{2 x - 2 = y - 3 x = - 9

DivEqn

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

{2 x - 2 = y x = 3

Now that we have found

x,

we can substitute it in Equation (I) to find

y .

{2 x - 2 = y x = 3 (I) (II)

Substitute

$(I):$ $x = 3$

{2 (3) - 2 = y x = 3

▼

(I):

Solve for

y

Multiply

$(I):$ Multiply

{6 - 2 = y x = 3

SubTerm

$(I):$ Subtract term

{4 = y x = 3

RearrangeEqn

$(I):$ Rearrange equation

{y = 4 x = 3

Exercise 53 Page 467

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