Answer:
The two lines are perpendicular as the multiplication of the two slopes of corresponding equations is -1.
- [tex]\frac{1}{2}\times-2=-1[/tex]
- [tex]m_1 \times m_2 = - 1[/tex]
Step-by-step explanation:
As the given equations
[tex]2x+y=4[/tex] and [tex]y=1/2x+4[/tex]
We have to determine the relationship between lines.
Let us consider
[tex]2x+y=4[/tex].....[A]
[tex]y=1/2x+4[/tex].....[B]
As we know that slope intercept form of an equation is
[tex]y=mx+c[/tex]
Here, m is the slope of the equation.
Compare [tex]y=1/2x+4[/tex] with [tex]y=mx+c[/tex]
Let us consider m₁ be the slope of [tex]y=1/2x+4[/tex]
[tex]y=1/2x+4[/tex] has a slope [tex]m_{1}=\frac{1}{2}[/tex]
Solving Equation [A]
[tex]2x+y=4[/tex]
[tex]y=-2x+4[/tex]......[C]
Compare [tex]y=-2x+4[/tex] with [tex]y=mx+c[/tex]
Let us consider m₂ be the slope of [tex]y=-2x+4[/tex]
[tex]y=-2x+4[/tex] has a slope [tex]m_{2}=-2[/tex]
Two lines will be perpendicular if the multiplication of the two slopes of corresponding equations is -1.
i.e. [tex]m_1 \times m_2 = - 1[/tex]
As
- [tex]y=1/2x+4[/tex] has a slope [tex]m_{1}=\frac{1}{2}[/tex]
- [tex]y=-2x+4[/tex] has a slope [tex]m_{2}=-2[/tex]
So, lets multiply the slopes of equations [A] and [B].
[tex]\frac{1}{2}\times-2=-1[/tex]
[tex]m_1 \times m_2 = - 1[/tex]
Therefore, the two lines are perpendicular. Please also check the attached figure to visualize the relationship.
Keywords: lines, perpendicular lines, slope
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