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The addition method, also known as the elimination method, is one approach to solving systems of linear equations. In this method, equations are manipulated so that adding them together will eliminate one of the variables.

For example, consider two equations:

• Equation 1: 3x - 4 = y + 18
• Equation 2: 4x + 5y = -21

The goal is to add or subtract one equation from the other to remove a variable. However, in the given equations, the coefficients of x and y are not opposites. This indicates that an adjustment is necessary. One could multiply each equation by an appropriate number to make the coefficients of one variable opposites. Conversely, you can also solve for one variable in terms of the other, as was done in Step 1 of the example, and then proceed with substitution, detailed in the following section.

Algebraic substitution is another technique for solving systems of equations. This method involves solving one equation for one variable and then substituting this expression into the other equation.

In the original exercise, the first equation is rearranged to isolate y (y = 3x - 22). This expression for y is then substituted into the second equation in place of y:

• 4x + 5(3x - 22) = -21

By substituting, you combine the two equations into one equation with one variable, making it possible to solve for x. After finding x, you substitute the value back into one of the original equations to find y, completing the system. If the substitution simplifies correctly and both variables are isolated successfully without any contradictions, the system has a single unique solution.

Linear equations form the foundation of algebra and provide relationships between variables in a straight line when graphed. They typically look like ax + by = c, where a, b, and c are constants.

A system of linear equations consists of two or more equations that share common variables. The solution to such a system is the point or points where the equations intersect on a graph.

In the context of our example, the intersection point corresponds to the values of x and y that satisfy both equations simultaneously. The final solution (x=4.68, y=-6.96) represents the point where the lines, if graphed, would cross. It's important to note that while graphing is a valuable visualization tool, in many cases, algebraic methods like the addition method and substitution provide a more precise and sometimes easier path to the same answer.