Simplify these expressions:
To simplify algebraic expressions, we identify and combine "like terms." Like terms are terms that contain the same variables raised to the same powers. For the first expression, \(4x + 7x + (-x)\), all terms contain the variable \(x\) raised to the power of 1. For the second expression, \(-5yz + yz + 2yz\), all terms contain the product of variables \(yz\). Once identified, we use the distributive property, which states that \(ax + bx + cx = (a + b + c)x\), to combine the coefficients of these like terms.
For the expression \(4x + 7x + (-x)\), we can identify the coefficients of \(x\). The terms are \(4x\), \(7x\), and \(-1x\) (since \(-x\) is equivalent to \(-1 \times x\)). Applying the distributive property, we add the coefficients:
\[4 + 7 - 1 = 10\]
Thus, the simplified form of the expression is \(10x\).
For the expression \(-5yz + yz + 2yz\), we identify the coefficients of the term \(yz\). The terms are \(-5yz\), \(1yz\) (since \(yz\) is equivalent to \(1 \times yz\)), and \(2yz\). Applying the distributive property, we add the coefficients:
\[-5 + 1 + 2 = -4 + 2 = -2\]
Thus, the simplified form of the expression is \(-2yz\).
The simplified expressions are:
In real life, combining like terms is essential for managing quantities. For example, if you are buying fruit, let \(a\) be the price of an apple and \(b\) be the price of a banana. If you buy 3 apples and 2 bananas, and then add 2 more apples and 1 more banana, your total expression is \(3a + 2b + 2a + 1b\). By combining like terms, you quickly find that you have \(5a + 3b\), which makes it much easier to calculate your total cost. Mastering this algebraic simplification allows you to organize data efficiently in budgeting, shopping, and project planning.