I Really Need Help, Pls Help. :)Which Properties Justify The Steps Taken To Solve The System?{2x4y=63x+2y=1Drag
When faced with solving systems of equations, understanding the properties that justify each step is crucial for both accuracy and clarity. The system given appears to be:
- 2x + 4y = 6
- 3x + 2y = 1
Before diving into the solution, it's important to recognize the properties involved in manipulating these equations. These include the properties of real numbers, the distributive property, the associative and commutative properties, and the properties related to equality such as addition and multiplication properties of equality. This article explores these properties in depth, illustrating how they justify each step taken to solve the system.
Understanding the System of Equations
Rewriting the Equations Clearly
Given the original equations:
- 2x + 4y = 6
- 3x + 2y = 1
Our goal is to find the values of x and y that satisfy both equations simultaneously. To do this, we often use algebraic methods like substitution or elimination, and each step relies on certain properties of numbers and equality.
Key Properties Justifying Steps in Solving the System
1. Addition and Subtraction Properties of Equality
The addition and subtraction properties of equality allow us to add or subtract the same quantity from both sides of an equation without altering its truth value. They are fundamental when manipulating equations to isolate variables.
- Addition Property of Equality: If a = b, then a + c = b + c.
- Subtraction Property of Equality: If a = b, then a - c = b - c.
2. Multiplication and Division Properties of Equality
Multiplying or dividing both sides of an equation by a non-zero number preserves equality, which is essential for solving for a variable.
- Multiplication Property of Equality: If a = b, then ac = bc, provided c ≠ 0.
- Division Property of Equality: If a = b and c ≠ 0, then a / c = b / c.
3. Distributive Property
The distributive property allows us to expand expressions like a(b + c) into ab + ac, or to factor expressions during simplification. It is often used to clear parentheses or to combine like terms.
- Distributive Property: a(b + c) = ab + ac.
4. Commutative Property
This property states that the order of numbers can be changed when adding or multiplying without affecting the result.
- Commutative Property of Addition: a + b = b + a.
- Commutative Property of Multiplication: ab = ba.
5. Associative Property
The associative property allows us to regroup terms when adding or multiplying without changing the result.
- Associative Property of Addition: (a + b) + c = a + (b + c).
- Associative Property of Multiplication: (ab)c = a(bc).
Applying These Properties Step-by-Step to Solve the System
Step 1: Simplify the Equations if Necessary
The first step involves ensuring the equations are in a standard form. In this case, the equations are already simplified:
- 2x + 4y = 6
- 3x + 2y = 1
Step 2: Decide on the Method — Elimination or Substitution
Choosing the elimination method is often straightforward here because the coefficients of y are similar after some adjustments. Alternatively, substitution might be used if one equation is easily solved for a variable.
Step 3: Use Multiplication to Align Coefficients (Elimination Method)
To eliminate one variable, we can manipulate the equations so that the coefficients of y are opposites. For example, multiply the first equation by 1 and the second by 2:
- Multiply equation 1 by 1 (no change): 2x + 4y = 6
- Multiply equation 2 by 2: 6x + 4y = 2
Justification: Using the multiplication property of equality, multiplying both sides of an equation by 1 or 2 preserves equality. This step prepares the equations for elimination by creating identical coefficients in y.
Step 4: Subtract Equations to Eliminate y
Subtract the first from the second:
(6x + 4y) - (2x + 4y) = 2 - 6
Applying the subtraction property of equality, we subtract corresponding sides:
- 6x - 2x = 4x
- 4y - 4y = 0
- 2 - 6 = -4
Resulting in:
4x = -4
Justification: The subtraction property of equality ensures the equation remains true when subtracting equal expressions from both sides.
Step 5: Solve for x
Divide both sides by 4:
x = -4 / 4 = -1
Justification: The division property of equality is used here; dividing both sides by 4 (a non-zero number) maintains the equality.
Step 6: Substitute x back into one of the original equations to find y
Using the second original equation:
3x + 2y = 1
3(-1) + 2y = 1
Simplify:
-3 + 2y = 1
Apply the addition property of equality to add 3 to both sides:
2y = 1 + 3 = 4
Finally, divide both sides by 2:
y = 4 / 2 = 2
Solution: x = -1 and y = 2.
Summary of Properties Used in the Solution
- Multiplication Property of Equality: Used to align coefficients for elimination.
- Subtraction Property of Equality: Used to eliminate a variable.
- Division Property of Equality: To solve for the variable after elimination.
- Addition Property of Equality: When isolating y after substitution.
Conclusion
In solving systems of linear equations, a clear understanding of the properties of real numbers and equality is vital. Each step in the elimination method relies on properties such as addition, subtraction, multiplication, and division properties of equality, as well as the distributive, commutative, and associative properties. Recognizing these properties not only justifies each algebraic manipulation but also ensures the process is logical, valid, and transparent.
Whenever you approach a system of equations, remember to think about these properties and how they enable you to perform necessary algebraic steps confidently. Mastering these concepts is essential for solving more complex systems and developing a strong foundation in algebra.
