In Exercises 19–30, solve each system by the addition method. 2x + 3y = 6 2x - 3y = 6

Verified step by step guidance

1. Write down the system of equations clearly: $\mathrm{2x}+\mathrm{3y}=6\mathrm{2x}-\mathrm{3y}=6$.
2. Add the two equations together to eliminate $y$ because the coefficients of $y$ are opposites (+3y and -3y). This gives: $(2x\; +\; 3y)\; +\; (2x\; -\; 3y)\; =\; 6\; +\; 6$.
3. Simplify the left side by combining like terms: $2x\; +\; 2x\; +\; 3y\; -\; 3y\; =\; 4x\; +\; 0\; =\; 4x$, and simplify the right side: $6\; +\; 6\; =\; 12$. So the equation becomes $4x\; =\; 12$.
4. Solve for $x$ by dividing both sides by 4: $x\; =\; \backslash (\backslash frac\{12\}\{4\}\backslash )$.
5. Substitute the value of $x$ back into one of the original equations to solve for $y$. For example, use $2x\; +\; 3y\; =\; 6$, plug in $x$, and solve for $y$.

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 values that satisfy all equations simultaneously. Understanding how to interpret and represent these systems is fundamental to solving them.

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 is effective when coefficients of one variable are opposites or can be made opposites by multiplication.

Solving for Variables After Elimination

Once a variable is eliminated, the resulting single-variable equation can be solved using basic algebra. After finding one variable, substitute it back into one of the original equations to find the other variable, completing the solution of the system.