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How to Solve the System of Linear Equations $\\begin{cases}2x + y = 8 \\\\ x - y = 1\\end{cases}$
Mathematics
Grade 8 (Junior High School)
Question Content
Solve the system of equations: $\\begin{cases}2x + y = 8 \\\\ x - y = 1\\end{cases}$
Correct Answer
$x=3$, $y=2$ (or written as $(3, 2)$)
Detailed Solution Steps
1
Step 1: Choose the elimination method, since the coefficients of $y$ in the two equations are 1 and -1, which are additive inverses. Add the two equations together: $(2x + y) + (x - y) = 8 + 1$
2
Step 2: Simplify the left side by combining like terms: $2x + x + y - y = 3x$, and the right side is 9. This gives the equation $3x = 9$
3
Step 3: Solve for $x$ by dividing both sides of $3x=9$ by 3: $x = 3$
4
Step 4: Substitute $x=3$ into the second original equation $x - y = 1$, getting $3 - y = 1$
5
Step 5: Solve for $y$: Rearrange $3 - y = 1$ to get $y = 3 - 1$, so $y=2$
6
Step 6: Verify by substituting $x=3$ and $y=2$ into both original equations: For $2x+y=8$, $2*3 + 2=6+2=8$ (true); for $x-y=1$, $3-2=1$ (true). The solution is valid.
Knowledge Points Involved
1
System of Linear Equations in Two Variables
A set of two linear equations with the same two variables (usually $x$ and $y$). The solution is an ordered pair $(x,y)$ that satisfies both equations simultaneously, representing the intersection point of the two lines when graphed.
2
Elimination Method for Solving Linear Systems
A method to solve systems of linear equations by adding or subtracting the equations to eliminate one variable. This works when the coefficients of one variable are equal or additive inverses, allowing the variable to cancel out, leaving an equation with only one variable to solve.
3
Substitution to Verify Solutions
After finding potential values for the variables, substitute them back into all original equations to confirm that each equation holds true. This ensures the solution is valid for the entire system, not just one equation.
4
Combining Like Terms
The process of adding or subtracting terms that have the same variables raised to the same powers. For example, $2x + x = 3x$ and $y - y = 0$, which is used to simplify equations during the solving process.
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