Problem 51 - Fundamentals of Mathematics

Question

For problems \(47-56\), simplify each expression by combining like terms. $$ -x+5 y-8 x-6 x+7 y $$

Step-by-Step Solution

Answer

The simplified expression is \(-15x + 12y\).

Step 1: Identify the Like Terms

Like terms are terms that have the same variable raised to the same power. In this expression, identify the terms with 'x' and the terms with 'y'. The terms with 'x' are \(-x, -8x, -6x\), and the terms with 'y' are \(+5y, +7y\).

Step 2: Combine the 'x' Terms

Combine the coefficients of the 'x' terms: \(-1 - 8 - 6 = -15\). Therefore, the 'x' terms simplify to \(-15x\).

Step 3: Combine the 'y' Terms

Combine the coefficients of the 'y' terms: \(+5 + 7 = 12\). Therefore, the 'y' terms simplify to \(+12y\).

Step 4: Write the Simplified Expression

Now, write the simplified expression using the combined terms from previous steps. The expression becomes:\(-15x + 12y\).

Key Concepts

Like Terms

When simplifying expressions in algebra, one important step is to identify "like terms." Like terms are terms within an expression that have the exact same variable components. This means they have the same variables, and these variables are raised to the same powers. For example, in the expression \(-x + 5y - 8x - 6x + 7y\), the like terms are the ones that include the variable \(x\) or \(y\).

• The terms \(-x\), \(-8x\), and \(-6x\) are like terms because they all involve the variable \(x\).
• Similarly, \(+5y\) and \(+7y\) are like terms because they both include the variable \(y\).

Once you have identified the like terms, you can proceed to combine them. This simplifies the expression, making it easier to work with. It's essential to remember that only like terms can be combined in this way.

Coefficients

After identifying like terms, the next step in simplifying an algebraic expression involves working with the "coefficients." A coefficient is the numerical part that directly multiplies the variable. For example, in the term \(-8x\), the coefficient is \(-8\). To combine like terms, you'll need to perform arithmetic operations on their coefficients.

• For the terms involving \(x\): The coefficients are \(-1\) for \(-x\), \(-8\) for \(-8x\), and \(-6\) for \(-6x\). Adding these together: \(-1 - 8 - 6 = -15\).
• For the terms involving \(y\): The coefficients are \(+5\) and \(+7\). Adding these together gives: \(+5 + 7 = +12\).

By accurately combining the coefficients, you simplify the expression in the most efficient way possible. This step doesn't change the variables themselves, only how much they are present in the expression.

Variable Terms

Understanding and working with "variable terms" is a crucial part of algebraic simplification. A variable term includes both a coefficient and a variable, such as \(-8x\), where \(-8\) is the coefficient and \(x\) is the variable.

• In our expression, \(-x\), \(-8x\), and \(-6x\) are all variable terms involving the variable \(x\).
• Similarly, \(+5y\) and \(+7y\) are variable terms with the variable \(y\).

When combining variable terms, you only alter their coefficients, keeping the variable part the same. Hence, after combining the coefficients of \(x\) terms, you get \(-15x\). For \(y\) terms, you obtain \(+12y\). The resulting simplified expression \(-15x + 12y\) clearly shows how variable terms are combined without altering their variables.
This understanding allows you to simplify expressions efficiently while maintaining their mathematical integrity.