Problem 27 - Algebra A Combined Function

Question

Solve each system of equations by the substitution method. $$ \left\\{\begin{array}{l} -x+2 y=10 \\ -2 x+3 y=18 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution is \(x = -6\) and \(y = 2\).

Step 1: Solve the First Equation for x

Start with the first equation \(-x + 2y = 10\). Rearrange to solve for \(x\):\[-x = 10 - 2y\]Multiply by \(-1\) to isolate \(x\):\[x = 2y - 10\]

Step 2: Substitute Expression for x into Second Equation

Use \(x = 2y - 10\) in the second equation \(-2x + 3y = 18\). Replace \(x\) with \(2y - 10\):\[-2(2y - 10) + 3y = 18\].

Step 3: Simplify and Solve for y

Distribute \(-2\) through the parenthesis:\[-4y + 20 + 3y = 18\].Combine like terms:\[-y + 20 = 18\].Subtract 20 from both sides:\[-y = -2\].Multiply by \(-1\) to isolate \(y\):\[y = 2\].

Step 4: Substitute y back to find x

Use \(y = 2\) and substitute back into the expression for \(x\):\[x = 2(2) - 10\].Calculate:\[x = 4 - 10\], so \(x = -6\).

Step 5: Check the Solution

Substitute \(x = -6\) and \(y = 2\) into both original equations to ensure they are valid.First equation: \(-(-6) + 2(2) = 10\) simplifies to \(6 + 4 = 10\), which is true.Second equation: \(-2(-6) + 3(2) = 18\) simplifies to \(12 + 6 = 18\), which is also true. The solution verifies correctly.

Key Concepts

System of Equations

A system of equations is a set of two or more equations with the same variables. In the given problem, we have two equations involving the variables \(x\) and \(y\). The goal is to find the values of these variables that satisfy both equations simultaneously. Solving a system of equations gives us a point that lies on the lines represented by these equations. This point is often an intersection if there is only one solution. In practical terms, systems of equations can model various real-world situations where multiple conditions need to be satisfied at the same time.

In our example, the system is:

• \(-x + 2y = 10\)
• \(-2x + 3y = 18\)

By solving these simultaneously using a method like substitution, you can effectively "work around" the equations to find the correct values for \(x\) and \(y\). It's crucial in both pure mathematics and applied fields like engineering and economics.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying equations to make them more solvable. In the context of solving a system of equations, it might require isolating variables or combining terms. This process is essential for methods like substitution, which heavily rely on clean manipulation of equations to substitute one variable with another expression.

For instance, we started with the equation:

• \(-x + 2y = 10\)

We then rearranged it to solve for \(x\), yielding:

• \(x = 2y - 10\)

Such algebraic manipulation made it possible to substitute \(x\) in the second equation, thus transforming a two-variable equation into one involving a single variable. Simplification, distribution, and combining like terms are standard actions often taken in these problems. Making these steps clear and straightforward is key to ensuring students can replicate them in different contexts.

Linear Equations

Linear equations are equations of the first degree, which means each term is either a constant or the product of a constant and a single variable. They graph as straight lines on the Cartesian plane. A linear equation in two variables typically takes the form \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants.

In our system of equations, both are linear:

• \(-x + 2y = 10\)
• \(-2x + 3y = 18\)

Linear equations can have one solution, zero solutions, or infinitely many solutions. A single solution involves the lines intersecting at one point. This solution set can be found using techniques like substitution, where one equation is used to express one variable in terms of another. Understanding the nature of linear equations as forming lines in a plane provides a helpful visualization for why substitution works and how solutions manifest.