Solve each system by elimination. In systems with fractions, first clear denominators.
4x + y = -23
x - 2y = -17

Verified step by step guidance

1. Write down the system of equations clearly: \[4x + y = -23\] \[x - 2y = -17\]
2. To use the elimination method, aim to eliminate one variable by making the coefficients of either \(x\) or \(y\) the same (or opposites) in both equations. Here, let's eliminate \(y\).
3. Multiply the second equation by 2 to match the coefficient of \(y\) in the first equation (since the first has \(+1y\) and the second has \(-2y\)): \[2(x - 2y) = 2(-17)\] which simplifies to \[2x - 4y = -34\]
4. Now, multiply the first equation by 4 to align the \(y\) coefficients with the second equation: \[4(4x + y) = 4(-23)\] which simplifies to \[16x + 4y = -92\]
5. Add the two new equations together to eliminate \(y\): \[ (16x + 4y) + (2x - 4y) = -92 + (-34) \] Simplify this to get an equation with only \(x\), then solve for \(x\).

Key Concepts

Systems 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 remaining variable. This technique often requires multiplying one or both equations by constants to align coefficients before elimination.

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.