Question

If \(a\) and \(b\) are integers, and \(2 a+b=17,\) then \(8 a+b\) cannot equal which of the following? a. -1 b. 33 c. 35 d. 65 e. 71

Short answer

The value that is not possible is 33.

Step-by-step solution

Express the given equation

Start with the given equation: ewline ewline ewline $$2a + b = 17$$

Solve for b in terms of a

Rearrange the equation to solve for b: ewline $$b = 17 - 2a.$$

Substitute b into the expression 8a + b

Substitute the expression for b into the expression 8a + b: ewline $$8a + b = 8a + (17 - 2a)$$

Simplify the expression

Combine like terms to simplify the expression: ewline $$8a + (17 - 2a) = 6a + 17.$$

Analyze possible values for the expression

Since 6a + 17 is the expression derived from 8a + b, evaluate which of the given answers isn't possible: ewline Which value, when set to equal 6a + 17, does not result in an integer solution for a?ewline Check each option:For a. -1: ewline $$6a + 17 = -1 \Rightarrow 6a = -18 \Rightarrow a = -3.$$ This value is valid. ewline For b. 33: ewline $$6a + 17 = 33 \Rightarrow 6a = 16 \Rightarrow a = \frac{8}{3}$$ This is not an integer value and is, therefore, not possible. For c. 35: $$6a + 17 = 35 \Rightarrow 6a = 18 \Rightarrow a = 3.$$ This value is valid. For d. 65: ewline $$6a + 17 = 65 \Rightarrow 6a = 48 \Rightarrow a = 8.$$ This value is valid. For e. 71: ewline $$6a + 17 = 71 \Rightarrow 6a = 54 \Rightarrow a = 9.$$ This value is valid.

Key concepts

Integer Solutions

A crucial part of many GMAT algebra problems is identifying integer solutions. These are solutions where the values are whole numbers, both positive or negative, including zero. In our problem, we start by expressing the equation given: \(2a + b = 17\). Our goal is to figure out which values of \(8a + b\) would lead to non-integer solutions for \(a\). When examining each option, we plug values back into the simplified form \(6a + 17 = y\), where \(y\) is one of the multiple-choice answers. If the result is that \(a\) is not an integer, then it is the answer we seek. Checking each step carefully is essential to ensure that we maintain the integrity of our integer requirement.

Linear Equations

A linear equation, such as the one in our problem \(2a + b = 17\), is an equation where each term is either a constant or the product of a constant and a single variable. These kinds of equations graph as straight lines on a coordinate plane. Our goal is to manipulate these equations to find relationships between variables. We often rearrange them to isolate a single variable, here solved as \(b\) in terms of \(a\): \(b = 17 - 2a\). This allows us to substitute back into another expression to simplify and solve.

Expression Simplification

Simplifying expressions helps to solve for unknown variables more straightforwardly. After we substitute \(b\) into the expression \(8a + b\), we get: \begingroup \(8a + (17 - 2a)\), \begingroup which simplifies to \(6a + 17\). By combining like terms, we reduce the complexity of the problem. Simplifying makes further calculations easier and helps us quickly identify solutions by inspection or substitution.

Substitution Method

The substitution method is a powerful tool in algebra, used to solve systems of equations or simplify expressions. Here, after having expressed \(b\) in terms of \(a\), we substitute \(b\) in the equation \(8a + b\) to derive our simplified form \(6a + 17\). This method is vital because it transforms complex problems into simpler ones. As seen in our example, we used substitution to test each multiple-choice value: we replaced \(8a + b\) with \(6a + 17\) and checked if \(a\) remained an integer. For instance, when substituting 33 for the expression: \(6a + 17 = 33\), solving gives \(a = \frac{8}{3}\). Since \(8/3\) is not an integer, this answer does not fit. This confirms that 33 is the value \(8a + b\) cannot equal, illustrating how substitution helps to verify solutions efficiently.