Problem 71 - Algebra 2

Question

Use the following system of equations \(\left\\{\begin{array}{c}{4 x-10 y=-3} \\\ {12 x+5 y=12}\end{array}\right.\) What is the value of \(x\) in the solution? Enter your answer as a fraction in simplest form.

Step-by-Step Solution

Answer

The value of x is \(\frac{3}{4}\).

Step 1: Multiply the second equation

To eliminate the variable y, multiply the second equation by 2 to make the coefficient of y in both equations the same in magnitude but opposite in sign. The system of equations will then become: \(4 x - 10 y = -3\) (first equation, unchanged) and \(24 x + 10 y = 24\) (second equation, multiplied by 2).

Step 2: Add the two equations

By adding the two equations, the variable y will cancel out because the coefficients are opposites. This will allow us to solve for x. Add the equations to get: \(4 x - 10 y + 24 x + 10 y = -3 + 24\), which simplifies to \(28 x = 21\).

Step 3: Solve for x

Divide both sides of the equation \(28 x = 21\) by 28 to solve for x: \(x = \frac{21}{28}\). The fraction can be simplified by dividing the numerator and the denominator by their greatest common divisor, which is 7. Therefore, \(x = \frac{21 ÷ 7}{28 ÷ 7} = \frac{3}{4}\).

Key Concepts

Linear Equations

Linear equations are the foundation of algebra and a fundamental part of understanding how to solve systems of equations. In its simplest form, a linear equation represents a straight line when plotted on a graph. These equations come in the format of \( ax + by = c \) where a, b, and c are constants with a and b not both zero.

To solve a system of linear equations, you need to find the values of the variables that satisfy all equations in the system simultaneously. This often involves methods such as substitution, graphing, and elimination, which we will discuss in the next sections. Simplifying these equations can include clearing fractions or decimals, and ensuring variables and numbers are on the appropriate sides of the equation to facilitate the solving process.

Elimination Method

The elimination method is a strategic way to solve a system of equations. The goal with this method is to eliminate one variable so that you can solve for the other. By either adding or subtracting the equations from each other, one variable is removed because the coefficients are set to cancel each other out.

Steps to Use the Elimination Method:

• Arrange the equations: Write both equations in standard form and ensure that the terms are lined up vertically.
• Eliminate a variable: Multiply one or both equations by certain numbers to get opposite coefficients for one variable. In our exercise, multiplying the second equation by 2 made the coefficient of y in the second equation positive 10, which is the additive inverse of -10 in the first equation.
• Add or subtract the equations: Perform the addition or subtraction to eliminate the variable. Here, adding the equations removed y, leaving us with an equation in one variable, x.
• Solve for the remaining variable: After eliminating one variable, solve the resulting single-variable equation for the remaining variable.
• Substitute to find the other variable: After finding one variable, use it to find the other one by substitution into any of the original equations, if necessary.

Algebraic Fractions

Algebraic fractions are simply fractions that contain variables in the numerator, the denominator, or both. They follow the same principles as arithmetic fractions but they also present the additional challenge of working with variables.

Key Points When Working with Algebraic Fractions:

• To add or subtract algebraic fractions, find a common denominator.
• To simplify an algebraic fraction, factor the numerator and denominator, and then divide out any common factors.
• To solve equations involving algebraic fractions, it can sometimes be useful to multiply every term by the least common denominator to clear the fractions.

In the exercise provided, after using the elimination method to solve for x, we end up with an algebraic fraction, \(x = \frac{21}{28}\). It's important to simplify fractions to make the answer more understandable and succinct. Here, both the numerator and the denominator are divisible by 7, giving us the simpler fraction \(x = \frac{3}{4}\), which is the solution to the original system of equations in its simplest form.