In Exercises 19–30, solve each system by the addition method. 3x = 4y + 1 3y = 1 - 4x

Verified step by step guidance

1. Rewrite both equations in standard form (Ax + By = C) to prepare for the addition method. For the first equation, subtract 4y and 1 from both sides to get $\mathrm{3x}-4y=1$. For the second equation, add 4x to both sides and subtract 1 from both sides to get $4x+3y=1$.
2. Align the two equations for addition: $3x\; -\; 4y\; =\; 1$ and $4x\; +\; 3y\; =\; 1$.
3. Multiply each equation by a suitable number so that the coefficients of either $x$ or $y$ are opposites. For example, multiply the first equation by 3 and the second equation by 4 to align the coefficients of $y$: $3(3x\; -\; 4y)\; =\; 3(1)$ and $4(4x\; +\; 3y)\; =\; 4(1)$.
4. Add the two resulting equations to eliminate $y$. This will give you an equation with only $x$. Solve this equation for $x$.
5. Substitute the value of $x$ back into one of the original equations to solve for $y$. This completes the solution to the system.

Key Concepts

System of Linear Equations

A system of linear equations consists of two or more linear equations with the same set of variables. The solution is the set of variable values that satisfy all equations simultaneously. Understanding how to interpret and manipulate these equations is essential for solving the system.

Addition Method (Elimination Method)

The addition method involves adding or subtracting equations to eliminate one variable, making it easier to solve for the remaining variable. This method requires aligning terms and possibly multiplying equations to create opposite coefficients for one variable.

Rearranging Equations

Before applying the addition method, equations often need to be rearranged into standard form (Ax + By = C). This makes it easier to identify coefficients and perform elimination. Rearranging involves moving all terms to one side and simplifying.