What Value of a Makes Two Linear Equations Parallel?

[ThomsonKevin]

Hello everyone, I was wondering if someone could solve this and explain it in detail. The answer is supposed to be a= -2. Let's cut to the chase:

If the two lines 2x + ay = 1 and ax + (a+4)y = 2 are parallel ,what is a?

 

[evinda]

Hello everyone, I was wondering if someone could solve this and explain it in detail. The answer is supposed to be a= -2. Let's cut to the chase:

If the two lines 2x + ay = 1 and ax + (a+4)y = 2 are parallel ,what is a?

Hello! (Wave)We can determine if two straight lines are parallel by checking their slopes.

Suppose that the equation for the first line is $y_1 = m_1x + b_1$ and the equation for the second line $y_2 = m_2x + b_2$.

For two lines to be parallel their slopes have to be same. That is, $m_1 = m_2$.

$$2x+ay=1 \Rightarrow ay=-2x+1 \overset{a \neq 0}{\Rightarrow} y=-\frac{2}{a}x+\frac{1}{a}$$

$$ax+(a+4)y=2 \Rightarrow (a+4)y=-ax+2 \overset{a \neq -4}{\Rightarrow} y=\frac{-a}{a+4}x+\frac{2}{a+4}$$So, it has to hold:

$$-\frac{2}{a}=\frac{-a}{a+4} \Rightarrow -2a-8=-a^2 \Rightarrow 2a+8=a^2 \Rightarrow a^2-2a-8=0$$Solve the equation $a^2-2a-8=0$ and you will find the possible values of $a$.

 

[Evgeny.Makarov]

You can also use the fact that the lines $A_1x+B_1y+C_1=0$ and $A_2x+B_2y+C_2=0$ are parallel iff $A_1/A_2=B_1/B_2\ne C_1/C_2$. Here if $A_2=0$, then $A_1/A_2=B_1/B_2$ means, by definition, that $A_1=0$, and similarly for $B_1$, $B_2$.

 

[ThomsonKevin]

Yes, exactly what I needed, thank you both of you!

 

[CellCoree]

Sure, I'd be happy to help solve this problem and explain the steps involved. First, let's start by defining what a linear function is. A linear function is a type of mathematical function that can be represented by a straight line on a graph. It follows the form y = mx + b, where m is the slope of the line and b is the y-intercept.

Now, let's look at the two given lines: 2x + ay = 1 and ax + (a+4)y = 2. We can rewrite these equations in the form y = mx + b by solving for y:

2x + ay = 1
ay = -2x + 1
y = (-2/a)x + 1/a

ax + (a+4)y = 2
(a+4)y = -ax + 2
y = (-a/a+4)x + 2/(a+4)

Notice that both equations have the same slope, -2/a. This means that the lines are parallel, since parallel lines have the same slope. Now, in order for the lines to be parallel, the y-intercepts must also be the same. This means that 1/a = 2/(a+4).

To solve for a, we can cross-multiply and simplify the equation:

1/a = 2/(a+4)
(a+4)(1/a) = (a+4)(2/(a+4))
a + 4 = 2
a = -2

Therefore, the value of a that makes the two lines parallel is -2. I hope this explanation was clear and helpful. Let me know if you have any further questions.