Question - Solving a Linear Equation with Multiple Terms
Solution:
First, expand the terms: \( 2(3x + 5) - 3(3x - 1) = 3(4 - x) \)
\( 6x + 10 - 9x + 3 = 12 - 3x \)
Simplify the equation: \( -3x + 13 = 12 - 3x \)
Since the \(x\)-terms are on both sides and they cancel each other out, we're left with:
\( 13 = 12 \)
This is clearly not true; therefore, there is no solution for this equation, and the lines represented by these equations would be parallel and never intersect.
Tutor Solution:
To solve the provided equation, we'll first distribute the coefficients inside the brackets.
The equation is:
\[2(3x + 5) - 3(3x - 1) = 3(4 - x)\]We distribute the 2 to both 3x and 5:
\[2 \cdot 3x + 2 \cdot 5 = 6x + 10\]Then, distribute the -3 to both 3x and -1:
\[-3 \cdot 3x -3 \cdot (-1) = -9x + 3\]Now, distribute the 3 on the right side of the equation:
\[3 \cdot 4 - 3 \cdot x = 12 - 3x\]After doing the distribution, the equation looks like this:
\[6x + 10 - 9x + 3 = 12 - 3x\]Now, combine like terms on the left side:
\[(6x - 9x) + (10 + 3) = 12 - 3x\]The left side simplifies to:
\[-3x + 13 = 12 - 3x\]Since the -3x term appears on both sides of the equation, they cancel each other out when you add 3x to both sides:
\[-3x + 3x + 13 = 12 - 3x + 3x\]Which simplifies to:
\[13 = 12\]This indicates that the equation is inconsistent because 13 will never equal 12, so there is no solution. This situation typically indicates that the lines represented by the original equations would be parallel and never intersect; thus, there are no values of x that would satisfy this equation. Consequently, the final answer is that the given equation has no solution.
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