Answer:
Step-by-step explanation:
This one is simple substitution... at least, substitution is the easiest method. The first equation is 3<em>x</em> – 30 = <em>y</em> and the second is 7<em>y</em> – 6 = 3<em>x</em>
As I look, I see 3 ways to use substitution to solve this:
- substitute 7<em>y</em> – 6 for 3<em>x</em>
- substitute 3<em>x</em> – 30 for <em>y</em>
- solve 3<em>x</em> – 30 = <em>y</em> for 3<em>x</em> and make it equal to 7<em>y</em> – 6
We're going to only use 1 method for the sake of time. Try the other two on your own. Assuming you don't make any mistakes, they will work.
<u>Method 1</u>:
3<em>x</em> – 30 = <em>y</em>
7<em>y</em> – 6 = 3<em>x</em> — initial system of equations
7<em>y</em> – 6 – 30 = <em>y</em> — substitute 7<em>y</em> – 6 for 3<em>x</em>
<u>7</u><u><em>y</em></u> – 6 – 30 = <u><em>y</em></u> — marking like terms, bold for constants, <u>underlined</u> for variables
7<em>y</em> – 36 = <em>y</em> — combining the constants and simplifying
Here, you could diverge into multiple paths: add 36 to both sides, subtract <em>y</em> from both sides, divide by 6 OR subtract 7<em>y</em> from both sides and divide by –6 . For the sake of time, I'm subtracting 7<em>y</em>, though I don't like dealing with negatives.
7<em>y</em> – 7<em>y</em> – 36 = <em>y</em> – 7<em>y</em> — subtract 7<em>y</em> from both sides
–36 = –6<em>y</em> — simplify
–36 ÷ –6 = –6<em>y</em> ÷ –6 — divide by –6 on both sides
<em>y</em> = 6 — simplify
Again, we can diverge here: substitute <em>y</em> into 3<em>x</em> – 30 = <em>y</em> or substitute <em>y</em> into 7<em>y</em> – 6 = 3<em>x</em>
I'm going to choose 3<em>x</em> – 30 = <em>y</em> but it will work either way, should you take the time (if you have it) to chase down every path this problem can take.
3<em>x</em> – 30 = <em>y</em> — initial equation
3<em>x</em> – 30 = 6 — substitute 6 for <em>y</em>
3<em>x</em> – 30 + 30 = 6 + 30 — add 30 to both sides to isolate 3<em>x</em>
3<em>x</em> = 36 — simplify the expression
3<em>x</em> ÷ 3 = 36 ÷ 3 — divide both sides by 3 to isolate <em>x</em>
<em>x</em> = 12 — simplify
So, we have <em>x</em> = 12 and <em>y</em> = 6 . We know they work for 3<em>x</em> – 30 = <em>y</em> but not if they work for 7<em>y</em> – 6 = 3<em>x</em> . Let's substitute those in to see if (12, 6) really is the solution point.
7<em>y</em> – 6 = 3<em>x</em> — original equation
7(6) – 6 ≟ 3(12) — substitute 6 for <em>y</em> and 12 for <em>x</em>
42 – 6 ≟ 36 — simplify by multiplying
36 = 36 ✔ — simplify by combining like terms on left side
Success! It works! We have found our solution!
I hope this helps increase your understanding of the concept. Have a great day!
