Chapter 3: Problem 75

\(\begin{aligned} 5 x-y &=1 \\ x-5 y &=-10 \end{aligned}\)

### Short Answer

Expert verified

(\frac{5}{8}, \frac{17}{8})

## Step by step solution

01

## Write down the system of equations

The given system of linear equations is: \( 5x - y = 1 \) \( x - 5y = -10 \)

02

## Solve for one variable in one equation

Let's solve the second equation for \(x\): \( x - 5y = -10 \) Adding \(5y\) to both sides gives: \( x = 5y - 10 \)

03

## Substitute the expression into the first equation

Substitute \( x = 5y - 10 \) into the first equation: \( 5(5y - 10) - y = 1 \)

04

## Simplify and solve for y

Simplify the substituted equation: \( 25y - 50 - y = 1 \) Combine like terms: \( 24y - 50 = 1 \) Add 50 to both sides: \( 24y = 51 \) Divide by 24: \( y = \frac{51}{24} = \frac{17}{8} \)

05

## Substitute the value of y to find x

Substitute \( y = \frac{17}{8} \) back into \( x = 5y - 10 \): \( x = 5 \times \frac{17}{8} - 10 \) \( x = \frac{85}{8} - 10 \) \( x = \frac{85}{8} - \frac{80}{8} \) \( x = \frac{5}{8} \)

06

## Write the solution as an ordered pair

The solution to the system of equations is: \( \left( \frac{5}{8}, \frac{17}{8} \right) \)

## Key Concepts

These are the key concepts you need to understand to accurately answer the question.

###### linear equations

Linear equations form the backbone of algebra and are vital in various real-life applications. A linear equation is an equation where the highest power of the variable(s) is one. For example, in the exercise provided, we have two linear equations: \(5x - y = 1\) and \(x - 5y = -10\). Both are in the form of \(Ax + By = C\), where \(A\), \(B\), and \(C\) are constants.

Linear equations can have one variable or multiple variables. When dealing with a system of linear equations, we aim to find the point(s) where the equations intersect. This means we need to find values of the variables that satisfy all equations simultaneously. Learning to solve these systems helps us understand how quantity relationships work in the real world.

Linear equations can have one variable or multiple variables. When dealing with a system of linear equations, we aim to find the point(s) where the equations intersect. This means we need to find values of the variables that satisfy all equations simultaneously. Learning to solve these systems helps us understand how quantity relationships work in the real world.

###### substitution method

The substitution method is a popular technique for solving systems of linear equations. It involves solving one equation for one variable and then substituting that expression into the other equation. This method simplifies the system from two equations in two variables to one equation in one variable.

In our exercise, we began by solving the second equation, \(x - 5y = -10\), for \(x\). Adding \(5y\) to both sides gave us \(x = 5y - 10\). Next, we substituted \(5y - 10\) for \(x\) in the first equation, \(5x - y = 1\), resulting in a solvable expression for \(y\).

When substituting, it’s essential to follow the algebraic steps carefully, simplifying and combining like terms systematically. This process ultimately leads to finding the values of both variables.

In our exercise, we began by solving the second equation, \(x - 5y = -10\), for \(x\). Adding \(5y\) to both sides gave us \(x = 5y - 10\). Next, we substituted \(5y - 10\) for \(x\) in the first equation, \(5x - y = 1\), resulting in a solvable expression for \(y\).

When substituting, it’s essential to follow the algebraic steps carefully, simplifying and combining like terms systematically. This process ultimately leads to finding the values of both variables.

###### ordered pairs

Once we solve a system of linear equations, we express the solution as an ordered pair, \((x, y)\). An ordered pair represents the coordinates of the point where the two lines (represented by our equations) intersect.

In our current exercise, after finding \(y\) through substitution and simplification, we substitute \(y\) back into one of the original equations to find \(x\). The final solution is \( ( \frac{5}{8}, \frac{17}{8} ) \). This ordered pair tells us that at these specific values of \(x\) and \(y\), both equations are satisfied simultaneously.

Understanding ordered pairs is crucial in graphing solutions on the coordinate plane, offering a visual interpretation of where equations intersect.

In our current exercise, after finding \(y\) through substitution and simplification, we substitute \(y\) back into one of the original equations to find \(x\). The final solution is \( ( \frac{5}{8}, \frac{17}{8} ) \). This ordered pair tells us that at these specific values of \(x\) and \(y\), both equations are satisfied simultaneously.

Understanding ordered pairs is crucial in graphing solutions on the coordinate plane, offering a visual interpretation of where equations intersect.