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Chapter 6: Problem 11

Solve each system by substitution. $$\begin{aligned}6 x-y &=5 \\\y &=x\end{aligned}$$

Short Answer

Expert verified
The solution is \((x, y) = (1, 1)\).

Step by step solution

01

Identify Substitution Opportunity

The given system is: \(6x - y = 5\) and \(y = x\). Notice that the second equation is already solved for \(y\), allowing us to substitute \(y\) in the first equation with \(x\).
02

Substitute to Solve for x

Substitute \(y = x\) into the first equation: \(6x - x = 5\). Simplify the equation to find \(x\).
03

Simplify the Equation

Simplify \(6x - x\) to get \(5x = 5\).
04

Solve for x

Divide both sides by 5 to isolate \(x\): \(x = \frac{5}{5} = 1\).
05

Substitute Back to Find y

Substitute \(x = 1\) back into \(y = x\) to find \(y\). Thus, \(y = 1\).
06

Verify Solution

Check the solution \((x, y) = (1, 1)\) by substituting into the original equations. For \(6x - y = 5\): \(6(1) - 1 = 5\) holds true. For \(y = x\): \(1 = 1\) also holds true.

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Key Concepts

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

Substitution Method
The substitution method is a powerful technique for solving systems of equations. It simplifies the process by allowing you to replace one variable with another. This method is particularly useful when one of the equations is already solved for a variable, as it is in the second equation of our exercise: \(y = x\). This direct relationship makes substitution straightforward. You can take the expression for \(y\) and replace it in the other equation.

Here鈥檚 a simple breakdown of this process:
  • Identify the equation that is solved for a variable. In this case, \(y = x\).
  • Substitute this expression into the other equation.
  • This reduces the system to a single equation with one variable.
By substituting \(y = x\) into \(6x - y = 5\), you essentially convert the problem into a simpler form: \(6x - x = 5\). This makes it much easier to solve.
Solving for Variables
Once you have simplified the system using substitution, the next step is to solve for the variables. In our exercise, the substitution resulted in a single equation: \(5x = 5\). Solving it involves isolating \(x\) on one side of the equation.

Here's how you can do it:
  • Simplify any expressions on both sides if needed. In our example, it is already simplified.
  • Divide both sides by the coefficient of \(x\), which is 5 in this case.
  • This gives you \(x = 1\).
With \(x\) found, substitute it back to find \(y\). Use the equation \(y = x\), which leads to \(y = 1\). This way, both variables are solved efficiently after substitution.
Verifying Solutions
Verification is a crucial step in solving systems of equations. It ensures that the solutions found fit both equations in the original system. After finding \(x = 1\) and \(y = 1\), substitute these values back into the original equations.

Here鈥檚 the verification process:
  • Check the solution in the first equation \(6x - y = 5\). Substitute \(x = 1\) and \(y = 1\) to see if the left-hand side equals 5. It does, as \(6(1) - 1 = 5\).
  • Check the second equation \(y = x\), where substituting values yields \(1 = 1\), which is true.
By verifying the solution in both equations, you can be confident that the solution \((x, y) = (1, 1)\) is correct.

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Most popular questions from this chapter

Solve each nonlinear system of equations analytically. $$\begin{aligned}&y=-x^{2}+2\\\&x-y=0\end{aligned}$$

A box has an open top, rectangular sides, and a square base. The volume of the box is 576 cubic inches, and the surface area of the outside of the box is 336 square inches. Find the dimensions of the box.

A manufacturer of refrigerators must ship at least 100 refrigerators to its two West Coast warehouses. Each warehouse holds a maximum of 100 refrigerators. Warehouse A holds 25 refrigerators already, while warehouse \(B\) has 20 on hand. It costs \(\$ 12\) to ship a refrigerator to warehouse \(\mathrm{A}\) and \(\$ 10\) to ship one to warehouse B. How many refrigerators should be shipped to each warehouse to minimize cost? What is the minimum cost?

A box has rectangular sides and a rectangular top and base that are twice as long as they are wide. The volume of the box is 588 cubic inches, and the surface area of the outside of the box is 448 square inches. Find the dimensions of the box.

Let \(A=\left[\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right], B=\left[\begin{array}{ll}b_{11} & b_{12} \\ b_{21} & b_{22}\end{array}\right],\) and \(C=\left[\begin{array}{ll}c_{11} & c_{12} \\\ c_{21} & c_{22}\end{array}\right]\) where all the elements are real numbers. Use these matrices to show that each statement is true for \(2 \times 2\) matrices. Let \(A=\left[\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right], B=\left[\begin{array}{ll}b_{11} & b_{12} \\ b_{21} & b_{22}\end{array}\right],\) and \(C=\left[\begin{array}{ll}c_{11} & c_{12} \\\ c_{21} & c_{22}\end{array}\right]\) where all the elements are real numbers. Use these matrices to show that each statement is true for \(2 \times 2\) matrices. \(c(A+B)=c A+c B,\) for any real number \(c\)
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