How do you solve linear equations with variables on both sides?

Combine like terms, isolate variable using inverse operations.

What are the concepts of no solution and infinitely many solutions in linear equations?

No solution occurs when simplifying leads to a false statement, while infinitely many solutions happen when both sides of the equation are identical.

Set up an equation based on the problem, simplify using algebraic techniques, and solve for the unknown variable.

Combining like terms simplifies the equation and aids in isolating the variable for solution.

Substitute the solution back into the original equation to confirm its validity.

When solving linear equations with variables on both sides, simplify each side by combining like terms or using the distributive property, then use inverse operations to isolate the variable on one side and constants on the other side.
Introduce the concepts of no solution and infinitely many solutions in linear equations, where a false statement indicates no solution and identical sides signify infinitely many solutions.

How do you solve linear equations with variables on both sides?
Combine like terms, isolate variable using inverse operations.
What are the concepts of no solution and infinitely many solutions in linear equations?
No solution occurs when simplifying leads to a false statement, while infinitely many solutions happen when both sides of the equation are identical.
How do you approach solving real-life problems involving linear equations?
Set up an equation based on the problem, simplify using algebraic techniques, and solve for the unknown variable.
What is the significance of combining like terms in linear equations?
Combining like terms simplifies the equation and aids in isolating the variable for solution.
How do you verify the solution to a linear equation?
Substitute the solution back into the original equation to confirm its validity.

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To solve linear equations with variables on both sides, simplify each side by combining like terms or using the distributive property.
Use inverse operations to gather variable terms on one side and constant terms on the other side to isolate the variable.
In an example with 10 - 4x = -9x, move the variable terms to one side by adding 4x to both sides, simplify to x = -2, and verify the solution.
In another example with 3(3x - 4) = 14x + 56, distribute on both sides, move variable terms using addition or subtraction, and simplify to x = 26.
Introduce the concepts of no solution and infinitely many solutions in linear equations.
In an example like 3(5x + 2) = 15x, if the equation simplifies to a false statement like 6 = 0, it indicates a no solution case.
For an equation like -8y - 2 = -8y - 2, where both sides are identical, it signifies infinitely many solutions.
In a real-life problem, a boat travels upstream and downstream for the same distance, with the downstream rate being x + 3 mph faster than the upstream rate.
By setting up an equation like 4x = 2.8(x + 3) for the boat's journey, simplify it by distributing and moving variable terms to one side.
Solve the equation to find x = 7, then calculate the distance traveled upstream as 28 miles, considering the boat's rate and time traveled.