Question

If \(2 a-6 b=2\) and \(3 b-3 a=9\), what is \(a b\) ? A. -4 B. 10 C. -2 D. \(-1 / 2\) E. -10

Short answer

B. 10

Step-by-step solution

Write the given equations

Write down the two given equations: 1) \(2a - 6b = 2\) 2) \(3b - 3a = 9\)

Simplify the equations

Simplify both equations. For equation (1), divide by 2: \(a - 3b = 1\). For equation (2), divide by 3: \(b - a = 3\)

Rearrange the second equation

Rearrange the second equation to express it in terms of \(a\): \(b = a + 3\)

Substitute into the first equation

Substitute \(b = a + 3\) into \(a - 3b = 1\): \(a - 3(a + 3) = 1\)

Solve for \(a\)

Solve the equation: \(a - 3a - 9 = 1\) \(-2a - 9 = 1\) \(-2a = 10\) \(a = -5\)

Solve for \(b\)

Use the value of \(a\) to find \(b\): \(b = a + 3 = -5 + 3 = -2\)

Calculate \(ab\)

Multiply \(a\) and \(b\) to find \(ab\): \(ab = (-5)(-2) = 10\)

Key concepts

linear equations

Linear equations are mathematical expressions that represent straight lines when graphed. They typically take the form of a = mx + bwhere 'm' is the slope and 'b' is the y-intercept. But when dealing with two variables like in our exercise, the equations are written differently. For example, we had:2a - 6b = 23b - 3a = 9.These equations can be simplified and represented in their simplest form using algebraic manipulation. Linear equations are crucial in mathematics because they appear in many different areas, from solving real-world problems to more advanced mathematical theories.

algebraic manipulation

Algebraic manipulation involves rearranging equations to make them easier to work with. In our example, we started with 2a - 6b = 2 and 3b - 3a = 9.We divided both sides of each equation by appropriate constants to simplify them. Our simplified forms were:a - 3b = 1andb - a = 3.This step helps to make the equations easier to handle and prepares us to use methods like substitution to find the variables. Manipulating equations involves basic operations like addition, subtraction, multiplication, and division.

solving for variables

Solving for variables means isolating one variable on one side of the equation. In the exercise, we needed to solve for 'a' and 'b'. First, we rewrote the second simplified equation b - a = 3as b = a + 3.Once we expressed 'b' in terms of 'a,' we substituted this back into the first simplified equation. This allowed us to solve for 'a' without dealing with 'b' directly. After finding 'a,' which was -5, we then used this value to solve for 'b.' Solving for variables often requires several steps and checks to ensure accuracy.

substitution method

The substitution method is one of the techniques used to solve systems of linear equations. It involves expressing one variable in terms of the other and substituting this expression into another equation. In our problem, after rearranging b - a = 3to get b = a + 3,we substituted it into the first simplified equation, a - 3b = 1.This substitution gave us a - 3(a + 3) = 1,which we further simplified to find 'a.' Once we had 'a,' we substituted it back to get 'b.' This method shows how one can reduce a problem with two variables to consecutive single-variable problems, simplifying the process significantly.