Solving Equations Determining Solution Types And Finding Solutions
In the realm of mathematics, equations serve as fundamental tools for expressing relationships between variables and constants. Solving equations involves finding the values of the variables that satisfy the given equality. This article delves into the process of solving equations, focusing on identifying the nature of solutions and determining their values. We will explore various scenarios, including equations with no solution, unique solutions, and infinite solutions, providing a comprehensive guide to understanding and tackling equation-solving challenges.
(a)
Analyzing the Equation and Determining the Solution Set
To solve this equation, we must first simplify both sides by applying the distributive property. This involves multiplying the constants outside the parentheses by the terms inside. On the left side, we have -3 multiplied by both y and 2, resulting in -3y - 6. On the right side, we have 2 multiplied by both y and 6, resulting in 2y + 12. Additionally, we have the constant term + 7 on the right side.
After applying the distributive property, the equation becomes:
-3y - 6 = 2y + 12 + 7
Next, we combine like terms on each side of the equation. On the right side, we can combine the constants 12 and 7, which gives us 19. The equation now reads:
-3y - 6 = 2y + 19
To isolate the variable y, we need to move all the terms containing y to one side of the equation and all the constant terms to the other side. We can achieve this by adding 3y to both sides and subtracting 19 from both sides. Adding 3y to both sides cancels out the -3y term on the left side and adds 3y to the right side, resulting in 5y on the right side. Subtracting 19 from both sides cancels out the 19 on the right side and subtracts 19 from the left side, resulting in -25 on the left side.
The equation now becomes:
-25 = 5y
Finally, to solve for y, we divide both sides of the equation by 5. Dividing -25 by 5 gives us -5, and dividing 5y by 5 gives us y. Therefore, the solution to the equation is:
y = -5
In this case, the equation has a unique solution, which is y = -5. This means that there is only one value of y that satisfies the given equation. We can verify this solution by substituting -5 for y in the original equation and checking if both sides are equal.
Substituting y = -5 into the original equation, we get:
-3(-5 + 2) = 2(-5 + 6) + 7
Simplifying both sides, we have:
-3(-3) = 2(1) + 7
9 = 2 + 7
9 = 9
Since both sides of the equation are equal, we can confirm that y = -5 is indeed the correct solution.
Conclusion for Equation (a)
The solution to the equation -3(y + 2) = 2(y + 6) + 7 is y = -5. This equation has a unique solution, indicating that there is only one value of y that satisfies the given equality. The process of solving this equation involved applying the distributive property, combining like terms, isolating the variable, and performing algebraic operations to arrive at the final solution.
(b)
Step-by-Step Solution and Explanation
To determine the solution for the equation 5(x + 1) - x = 4(x - 1) + 9, we need to follow a series of algebraic steps to isolate the variable x. This involves simplifying both sides of the equation, combining like terms, and performing operations to get x by itself.
First, we apply the distributive property to remove the parentheses on both sides of the equation. This means multiplying the numbers outside the parentheses by each term inside the parentheses.
On the left side, we have 5 multiplied by both x and 1, which gives us 5x + 5. Then, we subtract x from this result.
On the right side, we have 4 multiplied by both x and -1, which gives us 4x - 4. Then, we add 9 to this result.
After applying the distributive property, the equation becomes:
5x + 5 - x = 4x - 4 + 9
Next, we combine like terms on each side of the equation. Like terms are terms that have the same variable raised to the same power. On the left side, we can combine 5x and -x, which gives us 4x. On the right side, we can combine -4 and 9, which gives us 5. The equation now reads:
4x + 5 = 4x + 5
At this point, we notice that both sides of the equation are identical. This means that the equation is an identity, which is an equation that is true for all values of the variable. In other words, no matter what value we substitute for x, the equation will always hold true.
To further illustrate this, we can try to isolate the variable x. If we subtract 4x from both sides of the equation, we get:
5 = 5
This is a true statement, but it doesn't tell us anything about the value of x. It simply confirms that the equation is an identity.
Similarly, if we subtract 5 from both sides of the equation, we get:
4x = 4x
Again, this is a true statement, but it doesn't give us a specific value for x.
Because the equation is an identity, it has infinitely many solutions. Any real number can be substituted for x, and the equation will still be true. Therefore, the solution set for this equation is all real numbers.
Conclusion for Equation (b)
The equation 5(x + 1) - x = 4(x - 1) + 9 has all real numbers as solutions. This is because the equation simplifies to an identity, where both sides are equivalent regardless of the value of x. When an equation is an identity, it indicates that there are infinitely many solutions, and any real number will satisfy the equation. This contrasts with equations that have a unique solution or no solution, highlighting the diverse nature of equation solving in mathematics.
In summary, this article has provided a detailed explanation of how to solve equations and determine the nature of their solutions. By understanding the principles of algebraic manipulation and recognizing the characteristics of different types of equations, we can effectively solve a wide range of mathematical problems and gain a deeper appreciation for the power of equations in representing and solving real-world scenarios.
