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The substitution method is a technique for solving a system of linear equations.
The goal is to solve one of the equations for one variable, and then substitute that expression into the other equation.
This reduces the system of equations to a single equation with one variable, which you can then solve.

Here’s a step-by-step guide:

• Solve one of the equations for one of the variables.
  For example, if you have y = 3x + 2, y is already isolated.
• Substitute this expression into the other equation.
  This means replacing the y in the second equation with 3x + 2.
  This gives you a new equation with just one variable.
• Solve this new equation for the single variable (e.g., solve for x).
• Substitute back into the first equation to find the remaining variable (e.g., plug the value of x back in to find y).
• Write the solution as an ordered pair \((x, y)\).

In this exercise, the system was:
1) \(2x + 5y = -14\)
2) \(y = -2x + 2\)
We substituted \(y = -2x + 2\) into the first equation.
That gave a single equation with one variable, which we solved for x.
Then we substituted the value of x back in to find y.
This method is powerful because it transforms a problem with two variables into a simple problem with one.

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols.
In solving systems of linear equations using substitution, you need to be comfortable with basic algebraic operations.
These include:

• Substitution: Replacing one variable with an expression involving another variable.
• Simplification: Combining like terms and using distribution to make the equation simpler.
• Isolating variables: Using addition, subtraction, multiplication, and division to get the variable by itself on one side of the equation.

For instance, in the given problem:

• We substituted \(y = -2x + 2\) into \(2x + 5y = -14\), yielding:\[2x + 5(-2x + 2) = -14\]
• We then simplified step by step using distribution and combining like terms:
• \(2x + 5(-2x) + 5(2) = -14\)
• \(2x - 10x + 10 = -14\)
• \(-8x + 10 = -14\)
• Lastly, we isolated x by subtracting 10 from both sides and then dividing by -8:
  \(-8x = -24\)
• \(x = 3\)

Algebraic skills make it possible to translate real-world problems into mathematical problems and solve them logically.

A system of linear equations consists of two or more linear equations with the same variables.
The solution to the system is the point, or points, where the equations intersect.
This means finding the values for the variables that satisfy all equations simultaneously.

Here’s a quick overview:

• Graphically, each equation represents a straight line on a coordinate plane.
• The solution is the point where these lines intersect.
  In algebraic terms, this solution is represented as an ordered pair \((x, y)\).

For example, consider our system:
1) \(2x + 5y = -14\)
2) \(y = -2x + 2\)
We solved it by substitution, finding that the lines intersect at \((x, y) = (3, -4)\).
This point is where both equations are true.

Systems of linear equations are fundamental in many areas, such as physics, engineering, and economics.
They allow us to find unknown values and understand the relationships between different quantities effectively.