Solve each system by elimination. In systems with fractions, first clear denominators.
(2x-1)/3 + (y+2)/4 = 4
(x+3)/2 - (x-y)/2 = 3

Verified step by step guidance

1. Identify the system of equations: \(\frac{2x - 1}{3} + \frac{y + 2}{4} = 4\) and \(\frac{x + 3}{2} - \frac{x - y}{2} = 3\).
2. Clear the denominators in each equation by multiplying both sides by the least common multiple (LCM) of the denominators: - For the first equation, multiply both sides by 12 (LCM of 3 and 4). - For the second equation, multiply both sides by 2 (LCM of 2 and 2).
3. After clearing denominators, simplify each equation to get rid of fractions and write them in standard linear form \(Ax + By = C\).
4. Use the elimination method by aligning the two equations and adding or subtracting them to eliminate one variable. This may involve multiplying one or both equations by suitable constants to make the coefficients of one variable opposites.
5. Solve the resulting single-variable equation for that variable, then substitute back into one of the original simplified equations to find the value of the other variable.

Key Concepts

Clearing Denominators

Clearing denominators involves multiplying both sides of an equation by the least common denominator (LCD) to eliminate fractions. This simplifies the system into equations with integer coefficients, making them easier to manipulate and solve.

System of Linear Equations

A system of linear equations consists of two or more linear equations with the same variables. The goal is to find values for the variables that satisfy all equations simultaneously, often by methods such as substitution, elimination, or graphing.

Elimination Method

The elimination method solves systems by adding or subtracting equations to eliminate one variable, reducing the system to a single equation with one variable. This method is efficient when coefficients align to cancel variables easily.