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How to Solve the 2-Variable Linear System: x + 2y = 100 and x - y = 81
Mathematics
Grade 8 (Junior High School)
Question Content
Solve the system of linear equations: $\\begin{cases}x + 2y = 100 \\\\ x - y = 81\\end{cases}$
Correct Answer
$x=87\\frac{1}{3}, y=6\\frac{1}{3}$ or $x=\\frac{262}{3}, y=\\frac{19}{3}$
Detailed Solution Steps
1
Step 1: Label the equations for clarity: Equation (1): $x + 2y = 100$; Equation (2): $x - y = 81$
2
Step 2: Use the elimination method. Subtract Equation (2) from Equation (1): $(x + 2y) - (x - y) = 100 - 81$
3
Step 3: Simplify the left and right sides: $x + 2y - x + y = 19$, which simplifies to $3y = 19$
4
Step 4: Solve for $y$: $y = \\frac{19}{3} = 6\\frac{1}{3}$
5
Step 5: Substitute $y = \\frac{19}{3}$ into Equation (2): $x - \\frac{19}{3} = 81$
6
Step 6: Solve for $x$: $x = 81 + \\frac{19}{3} = \\frac{243}{3} + \\frac{19}{3} = \\frac{262}{3} = 87\\frac{1}{3}$
7
Step 7: Verify by substituting $x=\\frac{262}{3}$ and $y=\\frac{19}{3}$ back into both original equations to confirm they hold true.
Knowledge Points Involved
1
System of Linear Equations in Two Variables
A set of two linear equations that share two variables, where each equation describes a straight line on a coordinate plane. The solution is the point (x,y) that satisfies both equations, corresponding to the intersection of the two lines.
2
Elimination Method for Solving Linear Systems
A technique where you add or subtract the equations in a system to eliminate one variable, reducing the system to a single equation with one variable that can be solved directly. It is most effective when coefficients of one variable are the same or negatives of each other.
3
Substitution Method (Alternative)
Another common method where you solve one equation for one variable, then substitute that expression into the other equation to solve for the remaining variable. For this problem, you could solve Equation (2) for $x=x=81+y$, then substitute into Equation (1).
4
Fraction Arithmetic
The rules for adding, subtracting, multiplying, and dividing fractions, which are needed to solve for variables that result in non-integer values. This includes converting whole numbers to fractions with a common denominator to combine terms.
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