Problem 85 - Algebra 1

Question

Solve the linear system. $$\begin{array}{l}y=4 x \\\x+y=10\end{array}$$

Step-by-Step Solution

Answer

The solution to the given system of equations is \( x = 2 \) and \( y = 8 \).

Step 1: Substitution

Given the equations of the system, since \( y = 4x \) is already solved for y, we can substitute \( y \) in the second equation \( x + y = 10 \) with \( 4x \). Thus, the second equation after substitution will become \( x + 4x = 10 \).

Step 2: Solve for x

Now, simplify the equation and solve for \( x \). The equation simplifies to \( 5x = 10 \). Dividing both sides by 5, we obtain \( x = 2 \).

Step 3: Back substitution

Now, to find the value of \( y \), plug \( x = 2 \) back into the first equation \( y = 4x \). This gives us \( y = 4 * 2 \) which results in \( y = 8 \).

Key Concepts

Substitution Method

The substitution method is a fundamental technique used to solve systems of linear equations. It involves solving one of the equations for one variable and then substituting this expression into the other equation. This effectively reduces the system of equations to a single equation with one unknown, making it easier to solve.

In our example, since the first equation is already solved for \( y \) (i.e., \( y = 4x \)), we do not need any additional manipulation. We can directly substitute \( y \) in the second equation \( x + y = 10 \) with \( 4x \).

This changes the second equation to \( x + 4x = 10 \), simplifying the system by removing \( y \) from the equation. The substitution method is particularly useful when one of the equations in the system is easy to solve for one variable.

Solving Systems of Equations

Solving systems of equations involves finding the values of the variables that satisfy all equations in the system simultaneously. Linear equations are equations of the first degree, meaning they involve only terms that are linear with respect to each variable.

There are multiple methods to solve systems of equations, including graphing, substitution, and elimination. In our problem, the substitution method is used, but the essential goal remains to uncover the specific values for all the variables involved that make each equation true.

Once you have substituted and simplified the equations, the next step is to solve for the unknowns. This find the exact point where all equation graphs intersect, which gives the solution to the system. Here, by solving \( 5x = 10 \), we isolate and solve for \( x \).

• Fusion of equations simplifies them.
• Finding the single value for variables verifies the intersection point.

Algebraic Manipulation

Algebraic manipulation is a key skill in rearranging and simplifying equations to solve them effectively. Involves operations such as addition, subtraction, multiplication, and division, used to isolate variables and solve equations.

In our example, after substituting \( y = 4x \) into \( x + y = 10 \), the equation becomes \( x + 4x = 10 \). Here, algebraic manipulation is used to combine the \( x \) terms, \( x + 4x \), summing them to \( 5x \).

Next, by performing division, we further simplify \( 5x = 10 \) to find \( x = 2 \).

• Manipulating terms on both sides to single them out.
• Balancing the equation by keeping operations consistent.

This manipulation is crucial in simplifying equations and ensuring the solution process is straightforward and logical. It enables you to distill complex systems into manageable single-variable problems, which are easier to solve.