Problem 46 - Algebra A Combined Function

Question

Solve each system by the substitution method. First simplify each equation by combining like terms. $$ \left\\{\begin{array}{l} 5 x+2 y-4 x-2 y=2(2 y+6)-7 \\ 3(2 x-y)-4 x=1+9 \end{array}\right. $$

Step-by-Step Solution

Answer

x = 5, y = 0

Step 1: Simplify First Equation

Start with the first equation: \(5x + 2y - 4x - 2y = 2(2y + 6) - 7\). Combine like terms on the left side. This simplifies to \(x = 4y + 12 - 7\). Simplify further, yielding \(x = 4y + 5\).

Step 2: Simplify Second Equation

For the second equation: \(3(2x - y) - 4x = 1 + 9\). Expand and combine like terms. This becomes \(6x - 3y - 4x = 10\), which simplifies to \(2x - 3y = 10\).

Step 3: Substitute and Solve for x

Substitute \(x = 4y + 5\) from Step 1 into \(2x - 3y = 10\). This results in \(2(4y + 5) - 3y = 10\). Expand to get \(8y + 10 - 3y = 10\).

Step 4: Solve for y

Combine like terms in the equation \(8y + 10 - 3y = 10\) to get \(5y + 10 = 10\). Subtract 10 from both sides to obtain \(5y = 0\). Divide by 5 to find \(y = 0\).

Step 5: Solve for x

Substitute \(y = 0\) back into the simplified equation \(x = 4y + 5\). This becomes \(x = 4(0) + 5\), simplifying to \(x = 5\).

Step 6: Verify Solution

Substitute \(x = 5\) and \(y = 0\) back into the original equations to check. The first equation is \(5(5) + 2(0) - 4(5) - 2(0)\) simplifies to 5, matching the constant on the right. The second equation is \(3(2(5) - 0) - 4(5)\) simplifies to 10, matching the constant on the right. The solution \(x = 5\) and \(y = 0\) satisfies both equations.

Key Concepts

System of Equations

A system of equations is a collection of two or more equations that need to be solved together. Each equation in the system involves the same set of variables and you are tasked with finding values for those variables that satisfy all of the equations at the same time. Solving a system of equations can sometimes be a puzzle because the solution must fit *both* equations, not just one.
In our exercise, we have two equations:

• Equation 1: \(5x + 2y - 4x - 2y = 2(2y + 6) - 7\)
• Equation 2: \(3(2x - y) - 4x = 1 + 9\)

The goal is to solve them simultaneously. This often involves methods like substitution or elimination to reduce the system to a simpler problem that you can easily solve for one of the variables and use this result to find the other variable.

Like Terms

In algebra, like terms are terms that include the same variables raised to the same power. Only the coefficients (numbers in front of the variables) are different. Recognizing like terms is crucial because it allows you to combine them to simplify equations.
For instance, in our exercise, the expression \(5x + 2y - 4x - 2y\) contains like terms:

• \(5x\) and \(-4x\) are like terms because both involve the variable \(x\).
• \(2y\) and \(-2y\) are also like terms since they both involve \(y\).

Combining these like terms can simplify expressions significantly, which is the first step when solving system of equations. And once you have simpler expressions, it becomes much easier to work with the equations.

Simplification

Simplification in algebra helps to make complex expressions more manageable. It involves using basic mathematical operations to combine like terms, reduce fractions, and eliminate any unnecessary parts of equations.
The simplification process is applied directly in our exercise:

• In the first equation, combining \(5x\) and \(-4x\) gives \(x\). Similarly, \(2y\) combined with \(-2y\) results in 0, simplifying the equation to \(x = 4y + 5\).
• The second equation, when expanded and simplified from \(6x - 3y - 4x = 10\), reduces to \(2x - 3y = 10\).

This simplification process is crucial to make the substitution method feasible, as it provides simpler equations to substitute into and solve for the unknowns.

Algebraic Substitution

The substitution method is a powerful tool for solving systems of equations. It involves taking one of the equations and solving it for a single variable, then substituting that expression into the other equation.
In our exercise, we use substitution as follows:

• From the first simplified equation \(x = 4y + 5\), we substitute \(x\) in the second equation \(2x - 3y = 10\) to get \(2(4y + 5) - 3y = 10\).
• This substitution then allows us to solve for \(y\) independently, resulting in \(y = 0\).
• Substituting back \(y = 0\) into \(x = 4y + 5\) yields \(x = 5\).

Through this method, we effectively reduce a system of two equations to solving a single equation after substitution, simplifying our path to the answer.