Question

If \(2 x+3 y=8\) and \(2 x-2 y=2\), then \(x\) is: (A) 2 (B) \(2.75\) (C) \(2.2\) (D) \(2.25\) (E) \(2.5\)

Short answer

The solution for \( x \) is \( 2.2 \). Thus, the answer is (C).

Step-by-step solution

Write down the given equations

The first equation is: \[ 2x + 3y = 8 \]The second equation is: \[ 2x - 2y = 2 \]

Eliminate one of the variables

To eliminate one of the variables, subtract the second equation from the first equation: \[ (2x + 3y) - (2x - 2y) = 8 - 2 \]\[ 2x + 3y - 2x + 2y = 6 \]\[ 5y = 6 \]

Solve for y

Divide both sides of the equation by 5 to solve for \( y \): \[ y = \frac{6}{5} \] \[ y = 1.2 \]

Substitute y back into one of the original equations

Substitute \( y = 1.2 \) into the first equation: \[ 2x + 3(1.2) = 8 \]\[ 2x + 3.6 = 8 \]

Solve for x

Subtract 3.6 from both sides of the equation to isolate \( 2x \): \[ 2x = 8 - 3.6 \]\[ 2x = 4.4 \]Now divide by 2 to solve for \( x \): \[ x = \frac{4.4}{2} \]\[ x = 2.2 \]

Key concepts

algebra problem solving

Algebra problem solving often involves breaking down complex problems into simpler steps. In this exercise, you were given two equations and asked to solve for one variable. This involves logical thinking and systematic manipulation of equations. The process typically includes

1. writing down the given equations,
2. manipulating the equations to eliminate one of the variables,
3. solving for one variable,
and
4. back-substituting to find the value of the other variable.

Mastering these steps can help you solve a wide range of algebraic problems. Remember, practice is key to becoming proficient!

linear equations

Linear equations come in the form of \(ax + by = c\). In this exercise, you had two linear equations. Linear equations graph as straight lines and can be solved using various methods such as substitution, elimination, or graphical representation. The elimination method, used here, involves combining or subtracting equations to eliminate a variable. This simplifies the equation, making it easier to solve for the remaining variable. For instance, in this problem:

Subtracting the second equation from the first eliminated x, giving you a single equation in y.

Knowing how to manipulate linear equations is fundamental in mathematics and appears in various standardized tests, including the GMAT.

mathematics reasoning

Mathematics reasoning is all about understanding the logical flow of problem-solving. Here, reasoning skills helped in deciding to subtract the equations to eliminate a variable. It also guided the steps like substituting and isolating variables. Good mathematical reasoning involves:

• Recognizing patterns and relationships,
• Using logical steps and calculations,
• Applying prior knowledge to arrive at the solution.

Sharpening these skills helps in tackling more complex mathematics problems and is essential for success in exams like the GMAT.

GMAT preparation

GMAT preparation involves practicing different types of math problems, including linear equations and algebra. The GMAT tests your problem-solving, analytical, and reasoning skills through questions like the one above. Here are some tips to enhance your preparation:

• Regular practice: Solve a variety of problems to strengthen your understanding.
• Focus on weak areas: Identify and work on your weaknesses.
• Time management: Practice solving problems within a set time to improve speed.

By mastering these fundamental skills and strategies, you'll be better equipped to excel in the GMAT.