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Solve the System of Linear Equations Using Substitution: $y=6x-11$, $-2x-3y=-7$
Mathematics
Grade 8 (Junior High School)
Question Content
Solve using substitution, write solution as (x,y) no spaces. \n$y = 6x - 11$\n$-2x - 3y = -7$
Correct Answer
(2,1)
Detailed Solution Steps
1
Step 1: Substitute the expression for $y$ from the first equation $y=6x-11$ into the second equation. Replace $y$ in $-2x-3y=-7$ to get: $-2x - 3(6x - 11) = -7$
2
Step 2: Expand and simplify the equation: $-2x - 18x + 33 = -7$. Combine like terms: $-20x + 33 = -7$
3
Step 3: Isolate the variable $x$: Subtract 33 from both sides: $-20x = -7 - 33 = -40$. Then divide both sides by -20: $x = 2$
4
Step 4: Substitute $x=2$ back into the first equation $y=6x-11$: $y=6(2)-11=12-11=1$
5
Step 5: Write the solution in the required (x,y) no-spaces format: (2,1)
Knowledge Points Involved
1
Substitution Method for Systems of Linear Equations
A technique to solve a system of two linear equations with two variables by replacing one variable with an equivalent expression from the other equation, reducing the system to a single equation with one variable that can be solved directly. It is most useful when one equation already has a variable isolated (solved for that variable).
2
Combining Like Terms
The process of adding or subtracting terms that have the same variable raised to the same power. For example, $-2x - 18x$ combines to $-20x$, as both terms have the variable $x$ to the first power. This simplifies algebraic equations to make solving for variables easier.
3
Isolating a Variable
Using inverse operations (addition/subtraction, multiplication/division) to get a variable alone on one side of an equation. For example, to isolate $x$ in $-20x + 33 = -7$, we subtract 33 and divide by -20, which follows the property that equal operations performed on both sides of an equation keep the equation balanced.
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