Are These Points Solutions To 2x + Y = 12?

Dec 8, 2025 by Alex Johnson 43 views

Welcome, math enthusiasts! Today, we're diving into the fundamental concept of checking if specific points, represented as ordered pairs, satisfy a given linear equation. The equation we'll be working with is $2x + y = 12$ . Understanding this process is crucial as it forms the bedrock for graphing lines, solving systems of equations, and many other advanced mathematical topics. We'll systematically evaluate each ordered pair $(x, y)$ by substituting its values into the equation and seeing if the equality holds true. This isn't just about memorizing steps; it's about grasping the meaning of a solution – a point that makes the equation a true statement.

Understanding Solutions in Algebra

In the realm of algebra, a solution to an equation is essentially a value or a set of values for the variables that makes the equation true. For an equation with one variable, like $x + 5 = 10$ , the solution is $x = 5$ , because when you substitute 5 for $x$ , you get $5 + 5 = 10$ , which is a true statement. However, when we deal with equations that have two variables, such as our target equation $2x + y = 12$ , a solution isn't a single number but rather an ordered pair $(x, y)$ . This ordered pair represents a specific point on a coordinate plane. If substituting the $x$ -coordinate and the $y$ -coordinate from the ordered pair into the equation results in a true statement (the left side equals the right side), then that ordered pair is indeed a solution to the equation. If it results in a false statement, it means that particular point does not lie on the line represented by the equation.

Think of the equation $2x + y = 12$ as a rule that a specific set of points must follow. The ordered pair $(x, y)$ is a solution if and only if it follows this rule. The $x$ -value and the $y$ -value are intrinsically linked by this equation. When we test an ordered pair, we are essentially asking: "Does this specific location $(x, y)$ on the graph satisfy the condition defined by $2x + y = 12$ ?" This concept is fundamental because linear equations represent straight lines, and the solutions to the equation are precisely all the points that make up that line. So, checking if an ordered pair is a solution is equivalent to checking if that point lies on the line.

We'll be examining three specific ordered pairs: $(3,5)$ , $(6,0)$ , and $(0,12)$ . For each, we will isolate the $x$ and $y$ values and plug them into the equation $2x + y = 12$ . We'll perform the arithmetic carefully and then compare the result to the number 12. This methodical approach ensures accuracy and reinforces the underlying mathematical principle. It’s a straightforward process, but it requires attention to detail. Remember, the goal is to see if the equation balances out. Let's get started with the first pair!

Testing the Ordered Pair $(3,5)$

Let's begin our investigation with the first ordered pair: $(3,5)$ . In this pair, the $x$ -value is 3, and the $y$ -value is 5. Our equation is $2x + y = 12$ . To determine if $(3,5)$ is a solution, we need to substitute $x = 3$ and $y = 5$ into the equation and see if the statement remains true. So, we'll replace every instance of $x$ with 3 and every instance of $y$ with 5.

The equation becomes: $2(3) + 5 = 12$ .

Now, we perform the multiplication first, following the order of operations (PEMDAS/BODMAS): $2 imes 3 = 6$ .

So, the equation simplifies to: $6 + 5 = 12$ .

Next, we perform the addition: $6 + 5 = 11$ .

Our equation now reads: $11 = 12$ .

Is this statement true? No, $11$ is not equal to $12$ . Since the left side of the equation ( $11$ ) does not equal the right side ( $12$ ) after substituting the values from the ordered pair $(3,5)$ , we can definitively conclude that $(3,5)$ is not a solution to the equation $2x + y = 12$ . This means the point with coordinates $(3,5)$ does not lie on the line represented by this equation.

It's important to remember that just because one point isn't a solution doesn't mean no points are. We have two more pairs to check, and one of them might just make our equation sing! This process of substitution and verification is a fundamental skill in algebra, and practicing it with different points helps solidify your understanding of what it means for a pair of numbers to satisfy an equation. Keep this process in mind as we move on to the next ordered pair.

Evaluating the Ordered Pair $(6,0)$

Moving on, our second ordered pair to test is $(6,0)$ . Here, the $x$ -value is 6, and the $y$ -value is 0. We will substitute these values into our equation $2x + y = 12$ , just as we did before. The goal is to see if the substitution results in a true statement, where the left side equals the right side.

Substitute $x = 6$ and $y = 0$ into the equation: $2(6) + 0 = 12$ .

First, we handle the multiplication: $2 imes 6 = 12$ .

The equation simplifies to: $12 + 0 = 12$ .

Next, we perform the addition: $12 + 0 = 12$ .

Our equation now reads: $12 = 12$ .

Is this statement true? Yes, $12$ is indeed equal to $12$ . Because substituting the values from the ordered pair $(6,0)$ into the equation $2x + y = 12$ results in a true statement, we can confidently say that $(6,0)$ is a solution to the equation. This means the point $(6,0)$ lies perfectly on the line defined by $2x + y = 12$ .

Finding a solution feels great, doesn't it? It confirms that our understanding of substitution and equation balancing is on the right track. This point $(6,0)$ is one of the infinitely many points that constitute the line $2x + y = 12$ . We are one step closer to understanding the complete set of solutions. Now, let's tackle the final ordered pair and see if it also makes the equation true.

Verifying the Ordered Pair $(0,12)$

Finally, we arrive at our third and final ordered pair for today's analysis: $(0,12)$ . In this pair, the $x$ -value is 0, and the $y$ -value is 12. We will use the same substitution method to check if this ordered pair satisfies the equation $2x + y = 12$ .

Substitute $x = 0$ and $y = 12$ into the equation: $2(0) + 12 = 12$ .

Following the order of operations, we first perform the multiplication: $2 imes 0 = 0$ .

The equation simplifies to: $0 + 12 = 12$ .

Next, we perform the addition: $0 + 12 = 12$ .

Our equation now reads: $12 = 12$ .

Is this statement true? Absolutely! $12$ is equal to $12$ . Since substituting the values from the ordered pair $(0,12)$ into the equation $2x + y = 12$ yields a true statement, we can conclude that $(0,12)$ is also a solution to the equation. This signifies that the point $(0,12)$ lies on the line represented by $2x + y = 12$ .

So, out of the three ordered pairs we tested, $(6,0)$ and $(0,12)$ are solutions to the equation $2x + y = 12$ , while $(3,5)$ is not. This exercise beautifully illustrates how to verify solutions for linear equations with two variables. It’s a core skill that opens the door to understanding linear graphs and systems of equations. Remember, each true statement we found confirms a point that perfectly aligns with the rule defined by the equation. Keep practicing this skill, and you'll master algebraic solutions in no time!

Conclusion

We have successfully determined which of the given ordered pairs are solutions to the equation $2x + y = 12$ . By systematically substituting the $x$ and $y$ values from each ordered pair into the equation and checking if the resulting statement was true, we found that $(6,0)$ and $(0,12)$ are indeed solutions. The ordered pair $(3,5)$ , however, did not satisfy the equation, meaning it is not a point on the line represented by $2x + y = 12$ . This process of verification is fundamental in algebra, serving as a key step in understanding linear equations, graphing, and solving more complex mathematical problems. It reinforces the idea that a solution is a specific value or set of values that makes an equation hold true.

Mastering this technique allows you to identify points that lie on a line and understand the relationship between algebraic equations and their graphical representations. Keep practicing these substitutions with various equations and points. It’s a rewarding skill that builds confidence in your mathematical abilities.

For further exploration into linear equations and coordinate geometry, you might find the resources at Khan Academy and Math is Fun incredibly helpful. They offer a wealth of explanations, examples, and practice problems.