Solve each system by elimination. In systems with fractions, first clear denominators.
5x + 7y = 6
10x - 3y = 46

Verified step by step guidance

1. First, write down the system of equations clearly: \$5x + 7y = 6\( \)10x - 3y = 46$
2. To use the elimination method, we want to eliminate one variable by making the coefficients of either \(x\) or \(y\) the same (or opposites) in both equations. Notice that the coefficient of \(x\) in the second equation is \(10\), which is exactly twice the coefficient of \(x\) in the first equation (\(5\)).
3. Multiply the first equation by \(-2\) to make the coefficients of \(x\) opposites: \(-2(5x + 7y) = -2(6)\) which simplifies to \(-10x - 14y = -12\)
4. Now add this new equation to the second original equation: \((-10x - 14y) + (10x - 3y) = -12 + 46\) This will eliminate \(x\) and give you an equation with only \(y\).
5. Solve the resulting equation for \(y\). Once you find \(y\), substitute it back into one of the original equations to solve for \(x\).

Key Concepts

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. Understanding how to interpret and set up these systems is essential for solving them.

Elimination Method

The elimination method involves adding or subtracting equations to eliminate one variable, making it easier to solve for the other. This technique often requires multiplying one or both equations by constants to align coefficients before combining them.

Clearing Fractions

When equations contain fractions, multiplying both sides by the least common denominator removes the fractions, simplifying calculations. Clearing denominators helps avoid errors and makes the elimination process more straightforward.