Solving Equations How Much Less Is 3y Than 10

If x+2y=7 and 4x+11y=34, how much is 3y less than 10?

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In this article, we will delve into the step-by-step process of solving a system of linear equations and then using the solution to answer a specific question. The given system of equations is:

1. $x + 2y = 7$
2. $4x + 11y = 34$

Our goal is to determine by how much $3y$ is less than $10$. This requires us to first find the value of $y$ by solving the system of equations. We can use several methods to solve this system, including substitution, elimination, and matrix methods. In this case, we will use the elimination method, which involves manipulating the equations so that when they are added or subtracted, one of the variables is eliminated. This leaves us with a single equation in one variable, which we can easily solve.

Solving the System of Equations

Step 1: Multiply the First Equation by a Constant

To eliminate $x$, we can multiply the first equation by $-4$. This will give us a new equation where the coefficient of $x$ is the opposite of the coefficient of $x$ in the second equation. This makes it easy to eliminate $x$ when we add the equations together. So, multiplying the first equation by $-4$, we get:

$-4(x + 2y) = -4(7)$

This simplifies to:

$-4x - 8y = -28$

Now we have two equations:

1. $-4x - 8y = -28$
2. $4x + 11y = 34$

Step 2: Add the Modified Equation to the Second Equation

Next, we add the modified first equation to the second equation:

$(-4x - 8y) + (4x + 11y) = -28 + 34$

Combining like terms, we get:

$(-4x + 4x) + (-8y + 11y) = 6$

This simplifies to:

$3y = 6$

Step 3: Solve for $y$

Now, we can easily solve for $y$ by dividing both sides of the equation by $3$:

$y = \frac{6}{3}$

This gives us:

$y = 2$

So, the value of $y$ is $2$. Now we know the value of $y$, we can determine the value of $3y$ by multiplying $y$ by $3$.

Step 4: Substitute $y$ into the First Original Equation to Find $x$

Substitute $y = 2$ into the first original equation $x + 2y = 7$:

$x + 2(2) = 7$

This simplifies to:

$x + 4 = 7$

Subtract $4$ from both sides to solve for $x$:

$x = 7 - 4$

$x = 3$

Therefore, the value of $x$ is $3$.

Calculate $3y$

Now that we have found the value of $y$, we can calculate $3y$:

$3y = 3(2) = 6$

So, $3y$ equals $6$. The next step is to find by how much $3y$ is less than $10$. We can do this by subtracting $3y$ from $10$.

Finding How Much Less $3y$ is Than $10$

Step 5: Calculate the Difference Between $10$ and $3y$

To find by how much $3y$ is less than $10$, we subtract $3y$ from $10$:

$10 - 3y = 10 - 6 = 4$

Thus, $3y$ is $4$ less than $10$. This is the final answer to the problem.

Conclusion

In summary, we solved the system of equations $x + 2y = 7$ and $4x + 11y = 34$ to find the value of $y$. We used the elimination method, which involved multiplying one equation by a constant and then adding or subtracting the equations to eliminate one variable. Once we found the value of $y$, we calculated $3y$ and then determined by how much it was less than $10$. The result showed that $3y$ is $4$ less than $10$. This problem demonstrates the importance of understanding how to solve systems of equations and apply the solutions to answer specific questions. The ability to solve systems of equations is a fundamental skill in algebra and is used in many different areas of mathematics and science.

To accurately determine how much less 3y is than 10, it's imperative to first solve the system of equations presented. This involves finding the values of both x and y that satisfy both equations simultaneously. The system is as follows:

1. x + 2y = 7
2. 4x + 11y = 34

There are multiple methods to tackle this, including substitution and elimination. For this explanation, we will employ the elimination method. This approach aims to eliminate one variable, leaving us with a single equation in a single variable, which is much easier to solve.

To initiate the elimination process, we focus on making the coefficients of either x or y the same (but with opposite signs) in both equations. A practical approach here is to eliminate x. We can achieve this by multiplying the first equation by -4. This gives us a new equation:

-4(x + 2y) = -4(7)

Simplifying this, we get:

-4x - 8y = -28

Now we have a modified system of equations:

1. -4x - 8y = -28
2. 4x + 11y = 34

Next, we add these two equations together. This step is crucial as it eliminates x:

(-4x - 8y) + (4x + 11y) = -28 + 34

Combining like terms, we get:

(-4x + 4x) + (-8y + 11y) = 6

This simplifies to:

3y = 6

Now, we can easily solve for y by dividing both sides of the equation by 3:

y = 6 / 3

Therefore, y = 2. This is a significant milestone, as we have successfully found the value of y. Next, we can use this value to find x.

To find the value of x, we substitute y = 2 into one of the original equations. Let's use the first equation, x + 2y = 7. Substituting y = 2, we get:

x + 2(2) = 7

Simplifying, we have:

x + 4 = 7

To isolate x, we subtract 4 from both sides:

x = 7 - 4

Thus, x = 3. We now have the values for both x and y: x = 3 and y = 2. These values satisfy both original equations, confirming that our solution is correct.

After successfully solving the system of equations, which yielded the value of y as 2, the subsequent crucial step is to calculate the value of 3y. This calculation is a straightforward process, requiring us to multiply the value of y by 3. Given that y = 2, the calculation unfolds as follows:

3 * y = 3 * 2

Performing the multiplication, we find that:

3 * 2 = 6

Therefore, the value of 3y is 6. This value is pivotal as it forms the basis for the final part of the problem, which is to determine by how much 3y is less than 10. This involves comparing the calculated value of 3y with the number 10 to find the difference.

This step is essential in providing a clear answer to the original question. Now that we have the value of 3y, we can proceed to compare it with 10. The comparison will reveal the exact numerical difference, illustrating precisely how much smaller 3y is compared to 10. This comparison involves a simple subtraction, taking the value of 3y away from 10. This operation will provide us with the final numerical answer to the problem.

Having established that 3y equals 6, we now turn our attention to the central question: by how much is 3y less than 10? To answer this, we need to find the difference between 10 and the value of 3y, which we calculated as 6. This involves a simple subtraction operation:

10 - 3y = 10 - 6

Performing the subtraction, we get:

10 - 6 = 4

This result signifies that 3y is 4 less than 10. In other words, if we were to add 4 to the value of 3y (which is 6), we would arrive at 10. This difference of 4 provides a clear and concise answer to the original problem, quantifying the amount by which 3y falls short of 10.

This final calculation is a critical step in the problem-solving process. It not only provides the answer in numerical terms but also offers a clear understanding of the relationship between 3y and 10. The subtraction highlights the gap between the two values, allowing for a precise statement about their relative sizes. This step underscores the importance of accurate calculations and logical progression in mathematical problem-solving, leading to a definitive conclusion.

In conclusion, by systematically solving the system of equations, determining the value of 3y, and then comparing it with 10, we have successfully answered the question. The final answer, 4, represents the amount by which 3y is less than 10, providing a complete and satisfactory resolution to the problem.