In Exercises 19–30, solve each system by the addition method. 3x - 4y = 11 2x + 3y = - 4

Verified step by step guidance

1. Step 1: Write down the system of equations clearly: $\mathrm{3x}-4y=112x+3y=-4$.
2. Step 2: To use the addition method, 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 $y$: $3(3x\; -\; 4y)\; =\; 3(11)and4(2x\; +\; 3y)\; =\; 4(-4)$.
3. Step 3: After multiplication, the system becomes: $9x\; -\; 12y\; =\; 33and8x\; +\; 12y\; =\; -16$. Now add the two equations to eliminate $y$: $(9x\; -\; 12y)\; +\; (8x\; +\; 12y)\; =\; 33\; +\; (-16)$.
4. Step 4: Simplify the resulting equation to solve for $x$. Once you find $x$, substitute it back into one of the original equations to solve for $y$.
5. Step 5: Write the solution as an ordered pair $(x,\; y)$ representing the values of $x$ and $y$ that satisfy both equations.

Key Concepts

System of Linear Equations

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

Addition (Elimination) Method

The addition method involves adding or subtracting equations to eliminate one variable, making it easier to solve for the remaining variable. This requires manipulating the equations, often by multiplying by constants, to align coefficients for elimination.

Solving for Variables After Elimination

Once a variable is eliminated, the resulting single-variable equation is solved using basic algebra. After finding one variable, substitution back into one of the original equations allows solving for the other variable, completing the solution to the system.