Problem 8 - Algebra A Combined Function

Question

Solve each system of equations by the substitution method. $$ \begin{array}{l} y=5 x-3 \\ y=8 x+4 \end{array} $$

Step-by-Step Solution

Answer

The solution is \( x = -\frac{7}{3} \) and \( y = -\frac{44}{3} \).

Step 1: Understand the Substitution Method

The substitution method involves solving one of the equations for one variable, and then substituting this expression into the other equation. We already have both equations solved for \( y \).

Step 2: Set Equations Equal

Since both equations are equal to \( y \), we can set them equal to each other: \[5x - 3 = 8x + 4\]

Step 3: Solve for x

To isolate \( x \), first subtract \( 5x \) from both sides:\[-3 = 3x + 4\]Next, subtract \( 4 \) from both sides:\[-7 = 3x\]Finally, divide by 3:\[x = -\frac{7}{3}\]

Step 4: Substitute Back to Find y

Now that we know \( x = -\frac{7}{3} \), substitute this value into the first equation:\[y = 5\left(-\frac{7}{3}\right) - 3\]Calculate:\[y = -\frac{35}{3} - \frac{9}{3} = -\frac{44}{3}\]

Step 5: Write the Solution

The solution to the system of equations is \( x = -\frac{7}{3} \) and \( y = -\frac{44}{3} \). This means the lines intersect at this point.

Key Concepts

system of equations

When working with a system of equations, you're dealing with two or more equations that are interconnected through common variables. In the context of algebra, a system typically involves two equations with two variables. Solving a system of equations means finding the values for these variables that satisfy all given equations simultaneously.

There are several methods to solve these systems, and the substitution method is one of them. This method is particularly useful when one of the equations is already isolated for one variable. In our example:

• Equation 1: \( y = 5x - 3 \)
• Equation 2: \( y = 8x + 4 \)

Both equations directly express \( y \) in terms of \( x \). This means we can set them equal to one another to find the value of \( x \) first.

solving algebraic equations

Algebraic equations often have unknown values we need to solve for. Solving these involves a step-by-step process of manipulating the equations to isolate and solve for the unknowns. When using the substitution method, the key steps are:

• First, align the equations by expressing one variable in terms of the other. This step is often already done in formulating the system, particularly with linear equations.
• Substitute the expression for the isolated variable from one equation into the other equation. This reduces the two equations to one equation with one unknown.
• Solve the resulting equation to find the value of the unknown variable.
• Substitute the solution back into one of the original equations to find the value of the other variable.

In our example, after equating:\[5x - 3 = 8x + 4\]We solved for \( x \) by simplifying and isolating \( x \) on one side. Then, we substituted \( x \) back into one of the original equations to find \( y \). This ensures that the solution is consistent with both equations.

intersection of lines

In geometry, the intersection of lines refer to the point where two lines meet or cross each other. For linear equations, this point manifests as the solution to the system of equations representing the lines. The method we've used not only provides the algebraic solution, but also reveals the geometric relationship of these lines on a coordinate plane.

With our solutions \( x = -\frac{7}{3} \) and \( y = -\frac{44}{3} \), it means the lines described by our equations intersect at the point:

• \(-\frac{7}{3}, -\frac{44}{3}\)

This point represents the only set of values for \( x \) and \( y \) that satisfies both equations simultaneously. If plotted, both equations as lines on a graph would cross at this point providing a visual validation of our algebraic solution. This intersection is crucial for many applications, such as finding equilibrium points in economics or solutions in mixtures for chemistry.