Answer:
The first of the three consecutive even integers is zero.
Explanation:
The question asks us to find the first of three consecutive even integers and then describes another certain relationship between those unknown integers. Thus, the main concepts for understanding, setting up, and solving this problem are these:
Main concepts:
1. Consecutive even integers
2. Identifying other relationship given in the problem (turning the word problem into algebraic symbols and an equation)
3. Solving the equation(s)
4. Verifying the answer
1. Consecutive even integers
The question states that three consecutive even integers are involved (asking us to find the first of them).
Let's call these unknown integers x, y, and z (sequentially, where x is the first, y is the second, and z is the third).
Therefore, "x" is the unknown value that we ultimately need to find for this problem.
Given that these integers are consecutive even integers, and since each even integer is exactly 2 units apart from the next even integer, regardless of the value of x, y must be exactly 2 units larger than x. Algebraically, x + 2 = y.
Similarly, z must be exactly 2 units larger than y, so y + 2 = z.
2. Identifying other relationships given in the problem (turning the word problem into algebraic symbols/equations)
The problem states that "the sum of the first and second is 10 less than 3 times the third."
Recall that a "sum" is adding things together, so "the sum of the first and second" means x + y
The word "is" means "equals", so the expression x + y is equal to the rest of the sentence -- we just need to figure out what the expression for the other side of the equation is.
The phrase "10 less than" means "subtract 10 from whatever I talk about next".
The phrase "3 times the third" means 3 * z, sometimes just written 3z. So, "10 less than 3 times the third" means 3z - 10.
So, the phrase "the sum of the first and second is 10 less than 3 times the third" means x + y = 3z - 10
3. Solving the equation(s)
The three equations that we have from the problem are below:
x + 2 = y
y + 2 = z
x + y = 3z - 10
All three of these equations are true (at the same time). This is called a System of Equations.
Side Note: there are three letters, and three equations -- good -- If you have 3 letters, you need 3 or more equations to be able to solve the System of Equations.
A common way to solve a system of equations, and perhaps the most intuitive, is a method called "Substitution". For the Substitution method, one takes an expression equivalent to one of the unknowns and substitutes it into the place of that unknown in the other equations.
In our situation, we're trying to find "x", so we want to substitute all of the "y" and "z" with expressions having "x" in them.
So, using the first equation, x + 2 = y, "y" is equivalent to "x + 2", so we can substitute the expression "x + 2" in anywhere we see a "y". Doing so in the second and third equations gives:
(x+2) + 2 = z
x + (x+2) = 3z - 10
Observing that the second equation has "z" by itself, equivalent to the expression "(x+2) + 2", an expression with only "x", we can substitute "(x+2) + 2" everywhere we see a "z" in the last equation:
x + (x+2) = 3[(x+2) + 2] - 10
On the left-hand side, we can combine like-terms (the "x"s), and similarly, on the right-hand side, we can combine like-terms (the 2s), giving a slightly simplified equation:
2x+2 = 3[x+4] - 10
On the right-hand side, distributing the 3 gives:
2x+2 = 3x+12 - 10
On the right-hand side, combining like-terms gives:
2x+2 = 3x+2
Subtracting 2 from both sides gives:
2x = 3x
Subtracting 2x from both sides gives:
0 = x
So, this means that the first of the three consecutive integers is zero.
4. Verify the answer
If zero is the first of "three consecutive even integers", then the three consecutive even integers are 0, 2, and 4.
The "sum of the first and second" is 0 + 2, which is 2.
"10 less than 3 times the third" is 3*(4) - 10, which is 12-10, or just 2.
Since 2 equals 2, the equation/relationship is true. Therefore, zero is the first of the three consecutive even integers, as requested.