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Solve Simultaneous Linear Equations 2y+x=10 and y+x=8 Using Elimination Method
Mathematics
Grade 8 (Junior High School)
Question Content
Solve the simultaneous equations below using elimination: 2y + x = 10; y + x = 8
Correct Answer
x = 6, y = 2
Detailed Solution Steps
1
Step 1: Label the equations for clarity: Equation 1: $2y + x = 10$; Equation 2: $y + x = 8$
2
Step 2: Subtract Equation 2 from Equation 1 to eliminate the $x$ variable: $(2y + x) - (y + x) = 10 - 8$
3
Step 3: Simplify the left and right sides: $2y + x - y - x = 2$, which simplifies to $y = 2$
4
Step 4: Substitute $y = 2$ into Equation 2: $2 + x = 8$
5
Step 5: Solve for $x$: $x = 8 - 2 = 6$
6
Step 6: Verify by substituting $x=6$ and $y=2$ into both original equations to confirm they hold true.
Knowledge Points Involved
1
Elimination Method for Simultaneous Linear Equations
A technique to solve systems of linear equations by adding or subtracting the equations to eliminate one variable, reducing the system to a single-variable equation that can be solved directly. It is used when the coefficients of one variable are the same or can be made the same via multiplication.
2
Substitution of Known Values
After solving for one variable in a system of equations, substituting that known value back into one of the original equations to solve for the remaining unknown variable. This is a standard follow-up step in elimination and substitution methods for simultaneous equations.
3
Verification of Simultaneous Equation Solutions
The process of plugging the solved values of variables back into all original equations to ensure both equations are satisfied. This confirms that the solution is valid for the entire system, not just a single equation.
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