Create an equivalent system of equations using the sum of the system and the first equation. x + 4y = 8 4x + y = 2
1 answer:
Answer:
Equivalent systems of equations review
Step-by-step explanation:
We're given two systems of equations and asked if they're equivalent.
x + 4y = 8 (1)
4x + y = 2 (2)
Interestingly, if we sum the equations in System A, we get:
Replacing the first equation in System A with this new equation, we get a system that's equivalent to System A:
This is System B, which means that System A is equivalent to System B.
You might be interested in
I'll show you how to do one of the equations and one of the inequalities. All the others are done exactly in the same way: you'll only have to change the numbers, and it will be a good exercise.
Equations
Let's take the first equality as an example: we have
By definition, the absolute value of a number is the positive version of that number: if the number is already positive the absolute value doesn't change it; if a number is negative the absolute value changes its sign.
So, if the absolute value of a number is 5, than that number was already 5, or it was -5, and the absolute value changed it to positive 5.
So, the solutions are given by
Inequalities
Again, we'll use the first one as example. We have
By the same logic as before, the absolute value of a number is greater than 7 if the number is already greater than 7, or if it is smaller than -7. For example, we have |-10|=10>7.
So, we have
Instead, if we have an inequality with the "less than" sign, we have for example
Let's say . Let's find point
so that we can find
.
- Leah walks 40 yards south.
- Leah walks 60 yards west.
- Leah walks 10 yards north.
- Leah walks 20 yards east.
We have found that .
Now think about this scenario visually. We started at the center of something, which we call point , and then started moving around until we got to point
. We can then form line
between the points. However, realize that we can actually make a triangle. Just think of one of the legs as part of the x-axis and the other leg as part of the y-axis.
We can find the length of these parts, which is simply the absolute value of the coordinates of point . It may be a little hard to think about, but essentially, we can form a triangle with sides that consist of part of the x-axis, part of the y-axis, and
. We also know that the lengths of the legs are 40 and 30.
Since we are given the two lengths of the legs on the triangle and trying to find the length of the hypotenuse, we can use the Pythagorean Theorem. This states:
and
are the lengths two legs of the triangle
is the length of the hypotenuse
Thus, substituting in our values, we find:
The length of is 50.