Sure! To find a single equation that represents the line pair given by the equations [tex]\( x = 2y \)[/tex] and [tex]\( 2x = y \)[/tex], follow these steps:
1. Label the equations:
[tex]\[
\text{Equation 1:} \quad x = 2y
\][/tex]
[tex]\[
\text{Equation 2:} \quad 2x = y
\][/tex]
2. Express both equations in terms of one variable:
From Equation 1:
[tex]\[
x = 2y
\][/tex]
From Equation 2:
[tex]\[
2x = y
\][/tex]
3. Substitute the expression for [tex]\( x \)[/tex] from Equation 1 into Equation 2:
Substituting [tex]\( x = 2y \)[/tex] into [tex]\( 2x = y \)[/tex]:
[tex]\[
2(2y) = y
\][/tex]
Simplifying the left-hand side:
[tex]\[
4y = y
\][/tex]
4. Solve for [tex]\( y \)[/tex]:
[tex]\[
4y - y = 0
\][/tex]
[tex]\[
3y = 0
\][/tex]
[tex]\[
y = 0
\][/tex]
5. Substitute [tex]\( y = 0 \)[/tex] back into Equation 1 to find [tex]\( x \)[/tex]:
[tex]\[
x = 2(0)
\][/tex]
[tex]\[
x = 0
\][/tex]
6. Write the solution as an ordered pair:
[tex]\[
(x, y) = (0, 0)
\][/tex]
So, the single equation representing the line pair given by [tex]\( x = 2y \)[/tex] and [tex]\( 2x = y \)[/tex] has a common solution at the point [tex]\((0, 0)\)[/tex]. This shows that both lines intersect at the origin.
