Consider the system of linear equations below.
[tex]\begin{array}{r}
4 x-9 y=9 \\
-x+3 y=6
\end{array}[/tex]
Rewrite one of the two equations above in the form [tex]$a x+b y=c$[/tex], where [tex]$a, b$[/tex], and [tex]$c$[/tex] are constants, so that the sum of the new equation and the unchanged equation from the original system results in an equation in one variable.
Answer (2)
Multiply the second equation by 4 to prepare for eliminating x.
Obtain the new equation: − 4 x + 12 y = 24 .
Add the new equation to the first equation: ( 4 x − 9 y ) + ( − 4 x + 12 y ) = 9 + 24 , which simplifies to 3 y = 33 .
The rewritten equation is − 4 x + 12 y = 24 .
Explanation
Understanding the Problem We are given the system of equations:
4 x − 9 y = 9 − x + 3 y = 6
The goal is to rewrite one of the equations in the form a x + b y = c such that adding it to the other equation eliminates one variable.
Eliminating x Let's try to eliminate x . To do this, we can multiply the second equation by 4:
4 ( − x + 3 y ) = 4 ( 6 )
This simplifies to:
− 4 x + 12 y = 24
Now, we add this new equation to the first equation:
( 4 x − 9 y ) + ( − 4 x + 12 y ) = 9 + 24
Simplifying, we get:
3 y = 33
This is an equation in one variable, y .
Attempting to Eliminate y Alternatively, let's try to eliminate y . To do this, we can multiply the second equation by -3:
− 3 ( − x + 3 y ) = − 3 ( 6 )
This simplifies to:
3 x − 9 y = − 18
Now, we add this new equation to the first equation:
( 4 x − 9 y ) + ( 3 x − 9 y ) = 9 + ( − 18 )
Simplifying, we get:
7 x − 18 y = − 9
This does not eliminate a variable.
Final Answer Therefore, the correct approach is to multiply the second equation by 4 to eliminate x . The rewritten equation is:
− 4 x + 12 y = 24
This is in the form a x + b y = c , where a = − 4 , b = 12 , and c = 24 .
Examples
This concept is used in various fields like economics to solve systems of equations representing supply and demand, or in physics to analyze forces acting on an object. By eliminating one variable, we can simplify the problem and find the values of the remaining variables. For example, consider a scenario where you have two investment options with different interest rates and initial investments. By setting up a system of equations and eliminating one variable, you can determine the optimal investment strategy to maximize your returns.
To eliminate x from the system of equations, we multiplied the second equation by 4 yielding − 4 x + 12 y = 24 . Adding this new equation to the first results in an equation in one variable, 3 y = 33 . Thus, the rewritten equation is − 4 x + 12 y = 24 .
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