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Solve the System of Equations \( \\begin{cases} 2x + y = 8 \\\\ x - y = 1 \\end{cases} \)
Mathematics
Grade 7 (Junior High School)
Question Content
Solve the system of equations: \\( \\begin{cases} 2x + y = 8 \\\\ x - y = 1 \\end{cases} \\)
Correct Answer
The solution is \( x = 3 \), \( y = 2 \)
Detailed Solution Steps
1
Step 1: Add the two equations to eliminate \( y \). \n \( (2x + y) + (x - y) = 8 + 1 \)
2
Step 2: Simplify the equation. \n \( 3x = 9 \)
3
Step 3: Solve for \( x \). \n \( x = \\frac{9}{3} = 3 \)
4
Step 4: Substitute \( x = 3 \) into \( x - y = 1 \) to find \( y \). \n \( 3 - y = 1 \)
5
Step 5: Solve for \( y \). \n \( -y = 1 - 3 \\implies -y = -2 \\implies y = 2 \)
Knowledge Points Involved
1
System of Linear Equations in Two Variables
A set of two linear equations involving two variables (e.g., \( x \) and \( y \)), where a solution must satisfy both equations simultaneously.
2
Elimination Method
A technique to solve systems of equations by adding or subtracting equations to eliminate one variable, reducing the system to a single-variable equation that can be solved directly.
3
Substitution Method
After finding the value of one variable (e.g., \( x \)) from one equation, substitute this value into the other equation to solve for the remaining variable (e.g., \( y \)).
4
Solving Linear Equations
Using inverse operations (e.g., addition/subtraction, multiplication/division) to isolate the variable and find its value, based on the properties of equality.
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