Question

Solve and check each equation. Treat the constants in these equations as exact numbers. Leave your answers in fractional, rather than decimal, form. $$49-5 y=3 y-7$$

Short answer

The solution to the equation is \(y = 7\).

Step-by-step solution

Move all terms involving y to one side of the equation

Add 5y to both sides of the equation to move the terms involving y to one side. This eliminates the term -5y on the left. At the same time, add 7 to both sides to move the constant term to the other side. The equation becomes: \(49 + 7 = 3y + 5y\).

Simplify both sides of the equation

Combine like terms on both sides of the equation. On the left side, add 49 and 7 together. On the right side, combine the y terms by adding 3y and 5y. The equation simplifies to: \(56 = 8y\).

Solve for y

Divide both sides of the equation by 8 to solve for y. The equation becomes: \(y = \frac{56}{8}\).

Simplify the fraction

Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 8. The simplified fraction for y is: \(y = \frac{56 \div 8}{8 \div 8} = \frac{7}{1} = 7\).

Check the solution

Replace y with 7 in the original equation and verify that both sides are equal: \(49 - 5 \cdot 7 = 3 \cdot 7 - 7\). This simplifies to \(49 - 35 = 21 - 7\), which simplifies to \(14 = 14\), confirming that the solution y = 7 is correct.

Key concepts

Algebraic Equations

Algebraic equations are the cornerstone of mathematics, representing relationships between variables and constants. They come in various forms, but all share a common goal: to find the value(s) of the unknown variable(s) that makes the equation true.

Consider the equation \(49 - 5y = 3y - 7\)\. Here, \(y\) is our variable, and the numbers, 49 and -7 are constants. The equation sets two expressions equal to each other, and solving it involves finding the value of \(y\) that balances both sides. This balancing act is akin to solving a real-life puzzle where each move brings you closer to unveiling the hidden value of the variable.

When dealing with algebraic equations, always remember that what you do to one side, you must do to the other to maintain balance. This is the basic principle that will guide us as we manipulate and solve any algebraic equation.

Isolating Variables

Isolating variables is akin to finding a needle in a haystack. The goal is to get the variable we're solving for, on one side of the equation, all by itself. In the equation \(49 - 5y = 3y - 7\), we want to isolate \(y\).

To achieve this, we relocate all terms containing the variable to one side and all the constants to the other. This process often involves basic operations such as addition, subtraction, multiplication, or division, applied inversely to both sides of the equation. In our example, adding \(5y\) to both sides eliminates the \(y\)-term from the left and adding 7 to both sides removes the constant from the right. Ultimately, the variable stands alone, and that is when you know you're one step closer to deciphering its value.

Simplifying Expressions

Simplifying expressions is an essential stage in solving algebraic equations. It makes equations easier to handle by reducing them to their simplest form. You accomplish this by combining like terms and performing arithmetic operations where possible.

In our sample equation, \(56 = 8y\), the left side is a constant, but on the right side, we see the variable multiplied by a number. Simplifying expressions involves stripping down the equation to the bare essentials, making it easier to see the solution. Here, that simply means dividing both sides by 8 to isolate \(y\).

The act of simplifying can often reveal the solution as it presents the variables in an undressed, understandable form, allowing for a clearer path to the answer.

Equation Solving Steps

The process of solving equations is a step-by-step dance that must be followed meticulously to ensure accuracy. Each step plays a critical role in leading us to the solution. Breaking down our example problem into steps:

• Step 1: Move all \(y\)-terms to one side, and constants to the other.
• Step 2: Simplify by combining like terms on each side.
• Step 3: Isolate the variable by performing the inverse operation.
• Step 4: Simplify the fraction to find the value of the variable.
• Step 5: Verify the solution by substituting it back into the original equation.

Following these structured steps can transform a complex algebraic equation into a straightforward calculation, helping students build confidence in their problem-solving abilities. It's not just about finding an answer, but understanding how each move contributes to revealing the precise value of the variable in question.