Solving Linear Equations: Distributive Property

Questions and Answers

What is the best interpretation of the solution when a linear equation results in a variable expression set equal to the same variable expression?

The equation has infinite solutions.

How many solutions exist for the given equation: $\frac{1}{2}(x + 12) = 4x - 1$?

one

What is the value of x in the equation $\frac{1}{2}x (x - 14) + 11 = \frac{1}{2}x - (x - 4)$?

What is the value of x in the equation $1.5(x + 4) - 3 = 4.5(x - 2)$?

4 Signup and view all the answers

How many solutions exist for the given equation: $12x + 1 = 3(4x + 1) - 2$?

infinitely many Signup and view all the answers

Solve for x in the equation $9(x + 1) = 25 + x$.

x = 2 Signup and view all the answers

Which equation has no solution?

5 + 2(3 + 2x) = x + 3(x + 1) Signup and view all the answers

Solve for n in the equation $n + 1 = 4(n - 8)$.

n = 11 Signup and view all the answers

How many solutions exist for the equation: $0.75(x + 40) = 0.35(x + 20) + 0.35(x + 20)$?

one Signup and view all the answers

How many solutions exist for the equation: $3(x - 2) = 22 - x$?

one Signup and view all the answers

Solve for x in the equation $6(x - 1) = 9(x + 2)$.

x = -8 Signup and view all the answers

What is the best interpretation when the equation $\frac{2}{3}(6x - 3) = \frac{1}{2}(6x - 4)$ simplifies to $4x = 3x$?

The equation has one solution: x = 0. Signup and view all the answers

What is the value of x in the equation $2.5(6x - 4) = 10 + 4(1.5 + 0.5x)$?

2 Signup and view all the answers

How can Lily's partial solution from the equation $4(x - 1) - x = 3(x + 5) - 11$ be interpreted?

The equation has no solution. Signup and view all the answers

What is the value of n in the equation $\frac{1}{2}(n - 4) - 3 = 3 - (2n + 3)$?

2 Signup and view all the answers

How many solutions exist for the equation $3x + 13 = 3(x + 6) + 1$?

zero Signup and view all the answers

What is the value of x in the equation $1.5(x + 4) - 3 = 4.5(x - 2)$?

4 Signup and view all the answers

Solve for x in the equation $5(x - 10) = 30 - 15x$.

x = 4 Signup and view all the answers

Study Notes

Infinite Solutions

If a linear equation simplifies to the same variable expression on both sides, it has infinite solutions.

Unique Solutions

The equation ( \frac{1}{2}(x + 12) = 4x - 1 ) has one solution.
The equation ( 0.75(x + 40) = 0.35(x + 20) + 0.35(x + 20) ) results in one solution.
The equation ( 3(x - 2) = 22 - x ) also results in one solution.

Solution Equations

For the equation ( \frac{1}{2}x(x - 14) + 11 = \frac{1}{2}x - (x - 4) ), the solution leads to ( x = 0 ) after simplification.
The value of ( x ) in the equation ( 1.5(x + 4) - 3 = 4.5(x - 2) ) is found to be 4.
Solving ( 9(x + 1) = 25 + x ) results in ( x = 2 ).
( n + 1 = 4(n - 8) ) simplifies to ( n = 11 ).

No Solution

The equation ( 5 + 2(3 + 2x) = x + 3(x + 1) ) has no solution.
Interpretation of ( 4(x - 1) - x = 3(x + 5) - 11 ) indicates no solution.

Solutions Leading to Contradictions

The equation ( 6(x - 1) = 9(x + 2) ) simplifies to ( x = -8 ).
The equation ( 3x + 13 = 3(x + 6) + 1 ) yields zero solutions and indicates a contradiction.

Additional Key Values

For the equation ( 1.5(x + 4) - 3 = 4.5(x - 2) ), the solution for ( x ) is confirmed as 4 again.
In ( 5(x - 10) = 30 - 15x ), solving gives ( x = 4 ).

Summary of Solutions

Equations can lead to infinite solutions, unique solutions, or no solutions based on their structure and simplification. Ensure to analyze the equations for potential contradictions or consistent identities.

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