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Step-by-Step Solution for the System of Linear Equations $\\begin{cases}2x + 3y = 6 \\\\ 2y = 5 - x\\end{cases}$
Mathematics
Grade 8 (Junior High School)
Question Content
Solve the system of linear equations: $\\begin{cases}2x + 3y = 6 \\\\ 2y = 5 - x\\end{cases}$
Correct Answer
$x=-3, y=4$
Detailed Solution Steps
1
Step 1: Use the substitution method. Rearrange the second equation to solve for $x$: $x=5-2y$
2
Step 2: Substitute $x=5-2y$ into the first original equation $2x+3y=6$: $2(5-2y)+3y=6$
3
Step 3: Expand and simplify the equation: $10-4y+3y=6 \\Rightarrow 10-y=6$
4
Step 4: Solve for $y$: $-y=6-10=-4 \\Rightarrow y=4$
5
Step 5: Substitute $y=4$ back into $x=5-2y$: $x=5-2\\times4=5-8=-3$
6
Step 6: Verify by substituting $x=-3, y=4$ into the first equation: $2\\times(-3)+3\\times4=-6+12=6$, which matches the right-hand side, so the solution is valid.
Knowledge Points Involved
1
System of Linear Equations in Two Variables
A set of two linear equations with two unknown variables, where the solution is an ordered pair $(x,y)$ that satisfies both equations simultaneously. It is commonly used to model real-world scenarios with two related unknown quantities.
2
Substitution Method for Solving Linear Systems
An alternative method where you solve one equation for one variable in terms of the other, then substitute that expression into the second equation to get an equation with one variable, solve it, and substitute back. It is especially useful when one variable has a coefficient of 1 or -1.
3
Rearranging Linear Equations
The process of rewriting a linear equation to isolate one variable on one side of the equal sign, using inverse operations (addition/subtraction, multiplication/division). This is a foundational skill for substitution and graphing linear equations.
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