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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 $(3, 2)$
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: $8+1=9$. 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 equation $x - y = 1$, getting $3 - y = 1$
5
Step 5: Solve for $y$: Rearrange the equation to get $y=3-1=2$
6
Step 6: Verify the solution by substituting $x=3$ and $y=2$ into both original equations: For $2x+y=8$, $2*3+2=8$, which is true; for $x-y=1$, $3-2=1$, which is also true. The solution is valid.
Knowledge Points Involved
1
System of Linear Equations in Two Variables
A set of two linear equations that involve the same two variables, usually written in the form $\begin{cases} a_1x + b_1y = c_1 \\ a_2x + b_2y = c_2 \end{cases}$. 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 additive inverses (for addition) or equal (for subtraction), allowing us to solve for the remaining variable, then substitute back to find the eliminated variable.
3
Substitution of Variables
After solving for one variable in a system, substitute its value into one of the original equations to solve for the other unknown variable. This step is used to find the complete ordered pair solution of the system.
4
Verification of System Solutions
To confirm the validity of a solution, substitute the values of $x$ and $y$ back into both original equations of the system. If both equations hold true, the ordered pair is a valid solution.
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