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How to Solve the Linear System: 4(x+2)=1-5y and 3(y+2)=3-2x
Mathematics
Grade 8 (Junior High School)
Question Content
Solve the system of linear equations: $4(x+2)=1-5y$ and $3(y+2)=3-2x$
Correct Answer
x = -3, y = 1
Detailed Solution Steps
1
Step 1: Simplify both equations to the standard form $Ax + By = C$. For the first equation: Expand $4(x+2)=1-5y$ to get $4x + 8 = 1 - 5y$, then rearrange terms to $4x + 5y = 1 - 8$, so $4x + 5y = -7$.
2
Step 2: Simplify the second equation: Expand $3(y+2)=3-2x$ to get $3y + 6 = 3 - 2x$, then rearrange terms to $2x + 3y = 3 - 6$, so $2x + 3y = -3$.
3
Step 3: Use the elimination method. Multiply the simplified second equation by 2 to make the coefficients of $x$ match: $2*(2x + 3y) = 2*(-3)$ which gives $4x + 6y = -6$.
4
Step 4: Subtract the first simplified equation from the new equation: $(4x + 6y) - (4x + 5y) = -6 - (-7)$. This simplifies to $y = 1$.
5
Step 5: Substitute $y = 1$ into the simplified second equation $2x + 3y = -3$. We get $2x + 3*1 = -3$, so $2x = -3 - 3 = -6$, and $x = -3$.
6
Step 6: Verify the solution by plugging $x=-3$ and $y=1$ back into the original equations to confirm they hold true.
Knowledge Points Involved
1
Standard Form of Linear Equations
A linear equation in two variables is in standard form when written as $Ax + By = C$, where $A$, $B$, and $C$ are integers, and $A$ is non-negative. This form is useful for solving systems of equations using elimination or substitution methods.
2
Elimination Method for Linear Systems
The elimination method involves manipulating one or both equations in a system to create opposite coefficients for one variable, then adding or subtracting the equations to eliminate that variable, solving for the remaining variable, and substituting back to find the other variable. It is most efficient when coefficients can be easily matched or negated.
3
Distributive Property of Multiplication
The distributive property states that $a(b + c) = ab + ac$. It is used to expand expressions with parentheses, which is a necessary step to simplify linear equations before solving systems.
4
Verification of System Solutions
After finding values for variables in a system of equations, substituting the values back into the original equations confirms if they satisfy all equations in the system. A valid solution must make every equation in the system true.
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