Find The Solution: Equation $y-1=2(x+1/4)$

Hey guys! Welcome back to Plastik Magazine, where we break down those tricky math problems so you don't have to. Today, we're diving into a classic equation challenge: finding which ordered pair actually works in the equation $y-1=2 ext{ (x+1/4)}$. This might seem like a simple question, but mastering these kinds of problems is key to building a strong foundation in algebra. We'll walk through each option, show you the steps, and figure out which point is the true solution. So, grab your calculators (or just your brains!), and let's get solving!

Understanding the Equation and Ordered Pairs

Alright, let's kick things off by really understanding what we're dealing with here. The equation we've got is $y-1=2 ext{ (x+1/4)}$. This is a linear equation, meaning when you graph it, it'll form a straight line. The goal is to find an ordered pair (x, y) that makes this equation true. An ordered pair is just a pair of numbers, like $(x, y)$, where the first number is the x-coordinate and the second is the y-coordinate. Think of it like a specific point on a graph. For an ordered pair to be a solution to an equation, when you plug in the x and y values from that pair into the equation, both sides should be equal. If they aren't equal, then that point is not on the line represented by the equation.

Our equation, $y-1=2 ext{ (x+1/4)}$, involves a variable $y$ on one side and a variable $x$ on the other, with some constants and operations thrown in. Before we even start testing our options, it's super helpful to simplify the equation. This makes plugging in the numbers a lot easier and reduces the chances of making silly mistakes. Let's distribute the 2 on the right side: $2 imes x = 2x$ and 2 imes rac{1}{4} = rac{2}{4} = rac{1}{2}. So, the equation becomes y-1 = 2x + rac{1}{2}. Now, to get $y$ all by itself (which is often the easiest way to check solutions), we can add 1 to both sides: y = 2x + rac{1}{2} + 1. Since $1$ is the same as rac{2}{2}, we have y = 2x + rac{1}{2} + rac{2}{2}, which simplifies to y = 2x + rac{3}{2}. Or, if you prefer decimals, rac{3}{2} is $1.5$, so the equation is $y = 2x + 1.5$. This simplified form, $y = 2x + 1.5$, is going to be our best friend as we test each ordered pair. Remember, the key idea is substitution: swap out the $x$ and $y$ in our simplified equation with the values from the ordered pair and see if the equality holds true. If left side equals right side, bingo! You've found your solution. If not, you move on to the next pair.

Testing the Ordered Pairs: Step-by-Step

Now for the fun part, guys – putting our detective hats on and testing each ordered pair to see if it fits our equation, $y = 2x + 1.5$. We'll go through them one by one, plugging in the $x$ and $y$ values and doing the math.

Option (1): (0.75, 0)

Here, $x = 0.75$ and $y = 0$. Let's substitute these into $y = 2x + 1.5$:

$0 = 2(0.75) + 1.5$

First, calculate $2 imes 0.75$. That gives us $1.5$. So the equation becomes:

$0 = 1.5 + 1.5$

Which simplifies to:

$0 = 3$

Is $0$ equal to $3$? Nope, definitely not! So, ordered pair (0.75, 0) is not a solution. We can cross this one off the list.

Option (2): (1.25, 4)

For this pair, $x = 1.25$ and $y = 4$. Let's plug 'em in:

$4 = 2(1.25) + 1.5$

Calculate $2 imes 1.25$. That equals $2.5$. Now our equation looks like:

$4 = 2.5 + 1.5$

Add $2.5$ and $1.5$ together:

$4 = 4$

Well, look at that! $4$ is equal to $4$. This means that the ordered pair $(1.25, 4)$ is a solution to the equation. We've found our winner, but let's just quickly check the others to be absolutely sure and to practice our skills.

Option (3): (2.5, -6.5)

Here, $x = 2.5$ and $y = -6.5$. Substituting these values:

$-6.5 = 2(2.5) + 1.5$

First, calculate $2 imes 2.5$. That gives us $5$. The equation becomes:

$-6.5 = 5 + 1.5$

Add $5$ and $1.5$:

$-6.5 = 6.5$

Is $-6.5$ equal to $6.5$? Nope, they're opposites! So, $(2.5, -6.5)$ is not a solution.

Option (4): (4, -9.5)

Lastly, let's check $x = 4$ and $y = -9.5$:

$-9.5 = 2(4) + 1.5$

Calculate $2 imes 4$, which is $8$. The equation is now:

$-9.5 = 8 + 1.5$

Add $8$ and $1.5$:

$-9.5 = 9.5$

Again, $-9.5$ is not equal to $9.5$. So, this pair is also not a solution.

Conclusion: The Winning Ordered Pair

After meticulously checking each and every ordered pair, we've arrived at our answer. By substituting the $x$ and $y$ values from each option into our simplified equation, $y = 2x + 1.5$, we found that only one pair made the equation true. When we tested $(1.25, 4)$, we plugged in $x=1.25$ and $y=4$, and the equation $4 = 2(1.25) + 1.5$ resulted in $4 = 4$. This undeniable equality confirms that $(1.25, 4)$ is the correct solution. The other pairs, $(0.75,0)$, $(2.5,-6.5)$, and $(4,-9.5)$, all failed the test, resulting in false statements like $0=3$ or $-6.5=6.5$.

So, the ordered pair that is a solution to the equation $y-1=2 ext{ (x+1/4)}$ is (1.25, 4). It's awesome how a little bit of algebra can help us pinpoint the exact location of a point on a line. Remember, this method of testing points works for any linear equation. Just simplify the equation first, then substitute the x and y values from each ordered pair. If both sides match, you've found your solution. Keep practicing these skills, guys, and you'll be algebra wizards in no time! Stay tuned to Plastik Magazine for more math breakdowns and tips!