Solve the system of linear equations begin cases 2x y 8 x y...

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--- ## Question 1 **Question content**: Solve the system of linear equations: \(\begin{cases} 2x + y = 8 \\ x - y = 1 \end{cases}\) **Discipline**: Mathematics **Grade**: Seventh grade (junior high school, typically when systems of linear equations are introduced) ### Correct answer The solution to the system is \(\boldsymbol{\begin{cases} x = 3 \\ y = 2 \end{cases}}\) ### Detailed problem-solving steps 1. **Eliminate the \(y\)-variable by adding the two equations** The system is: \[ \begin{cases} 2x + y = 8 \quad (1) \\ x - y = 1 \quad \ \ (2) \end{cases} \] Add the left sides and right sides of equations \((1)\) and \((2)\): \[ (2x + y) + (x - y) = 8 + 1 \] Simplify the left side ( \(y - y = 0\) ) and the right side: \[ 3x = 9 \] 2. **Solve for \(x\)** Divide both sides of \(3x = 9\) by 3: \[ x = \frac{9}{3} = 3 \] 3. **Substitute \(x = 3\) into one of the original equations to solve for \(y\)** Use equation \((2)\): \(x - y = 1\). Substitute \(x = 3\): \[ 3 - y = 1 \] Subtract 3 from both sides: \[ -y = 1 - 3 = -2 \] Multiply both sides by \(-1\): \[ y = 2 \] 4. **Verify the solution** Substitute \(x = 3\) and \(y = 2\) into both original equations: - For \(2x + y = 8\): \(2(3) + 2 = 6 + 2 = 8\) (matches). - For \(x - y = 1\): \(3 - 2 = 1\) (matches). ### Knowledge points involved 1. **System of linear equations** - Definition: A set of two or more linear equations with the same variables (e.g., \(x\) and \(y\)) that are solved simultaneously. - Application: Used to model real-world problems with multiple unknowns (e.g., cost, distance, mixture problems). 2. **Elimination method** - Definition: A technique to solve a system of linear equations by adding or subtracting equations to eliminate one variable, reducing the system to one equation with one variable. - Key idea: Align coefficients of a variable (or their opposites) to cancel the variable when equations are combined. 3. **Substitution method (implicitly used in verification)** - Definition: Solve one equation for a variable, then substitute the expression into the other equation to reduce the system to one variable. - Application: Useful when one variable is easily isolated (e.g., \(y = 8 - 2x\) from the first equation). 4. **Verification of solutions** - Definition: Substitute the found values back into all original equations to confirm they satisfy every equation. - Importance: Ensures the solution is correct (avoids arithmetic errors). 5. **Solving linear equations (e.g., \(3x = 9\) or \(3 - y = 1\))** - Basic algebraic operations: Isolate the variable by inverse operations (e.g., division, subtraction, multiplication by \(-1\)). - Foundation: Essential for solving all types of equations in algebra.

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