What is the value of 𝑘 for which
the simultaneous equations 𝑥 plus two 𝑦 equals one and five 𝑥 plus 𝑘𝑦 equals
five do not have a unique solution?
Equations that do not have a unique
solution do not intersect, which means they are parallel lines and they have equals
slopes. To find the slope, let’s take both
of these equations and put them in the slope-intercept form. That’s the form 𝑦 equals 𝑚𝑥 plus
𝑏. And in this form, 𝑚 represents the
To get equation one in the
slope-intercept form, we need to isolate 𝑦. We can do that by subtracting 𝑥
from both sides of the equation. We then have two 𝑦 equals one
minus 𝑥. But because we’re trying to get the
form 𝑦 equals 𝑚𝑥 plus 𝑏, we can flip the 𝑥 and the 𝑏. It will then say two 𝑦 equals
negative 𝑥 plus one.
From there, we divide everything by
two. 𝑦 equals negative 𝑥 over two plus
one-half. But remember we’re looking for that
𝑚 value. So we wanted to say 𝑦 equals
negative one-half times 𝑥 plus one-half. 𝑚 equals the slope and it equals
We want to get our second equation
in the same form. So we subtract five 𝑥 from both
sides. 𝑘𝑦 equals five minus five 𝑥. We want our 𝑥 term first. So we’ll flip them around, being
careful to keep that negative with our 𝑥. 𝑘𝑦 equals negative five 𝑥 plus
After that, we want 𝑦 by
itself. So we divide everything by 𝑘. 𝑦 equals negative five over 𝑘𝑥
plus five over 𝑘. The slope of our second equation is
negative five over 𝑘. And we want to know what 𝑘 is. These two lines do not have a
unique solution. And that means they have equal
slopes. Negative one-half equals negative
five over 𝑘.
Let’s get the 𝑘 out of the
denominator, multiply by 𝑘 on both sides. 𝑘 times negative one equals
negative 𝑘. The denominator stays negative
two. And that equals negative five. To get two out of the denominator,
we multiply by two on both sides. Negative 𝑘 equals negative 10. And that means positive 𝑘 equals
When 𝑘 equals 10, these two lines
have the same slope. And that means they have no unique
solution. But before we leave this problem, I
want to show you one other way to think about it.
This time, we’re going to look at
the coefficients of 𝑥 and 𝑦 in both of these equations. And we’re going to make a ratio of
these coefficients. The coefficient of 𝑥 in the first
equation is one. We could say one 𝑥. And the coefficient of 𝑦 is
two. They have a ratio of one to
two. In the second equation, the
coefficient of 𝑥 is five and the coefficient of 𝑦 is 𝑘.
Notice that to get from one to two,
we multiply by two. And if we want these two ratios to
be equal, then to go from five to 𝑘, we will also multiply by two. Five times two is 10. So in order for these equations to
be parallel, 𝑘 must be equal to 10.
Both methods are equally valid. One just requires a little bit more
algebra than the other.