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Chapter 9: Problem 12

Solve each system by substitution. $$\begin{aligned} &4 x-5 y=-11\\\ &2 x+y=5 \end{aligned}$$

Short Answer

Expert verified
The solution is \(x = 1\) and \(y = 3\).

Step by step solution

01

Solve one equation for one variable

Start with the second equation, which is simpler: \[2x + y = 5\]Solve for y: \[y = 5 - 2x\]
02

Substitute solved value into the other equation

Now take the expression for y from Step 1 and substitute it into the first equation:\[4x - 5y = -11\]Substitute \(y = 5 - 2x\): \[4x - 5(5 - 2x) = -11\]
03

Simplify the equation

Distribute and simplify:\[4x - 25 + 10x = -11\]Combine like terms:\[14x - 25 = -11\]
04

Solve for x

Add 25 to both sides:\[14x = 14\]Divide by 14:\[x = 1\]
05

Solve for y using the value of x

Substitute \(x = 1\) back into the expression for y found in Step 1:\[y = 5 - 2(1)\]\[y = 3\]
06

Verify the solution

Substitute \(x = 1\) and \(y = 3\) back into the original equations to verify: First equation: \[4(1) - 5(3) = -11\]\[4 - 15 = -11\] True. Second equation: \[2(1) + 3 = 5\] \[2 + 3 = 5\]Also true. The solution is verified.

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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 technique to solve a system of linear equations. It involves solving one of the equations for one variable and then substituting that expression into the other equation. This reduces the system to a single equation with one unknown variable.

Here's a step-by-step approach:
  • Start by isolating one variable in one of the equations.
  • Next, substitute the isolated variable's expression into the other equation.
  • Simplify the second equation and solve for the remaining variable.
  • Finally, substitute this solution back into the expression obtained initially to find the value of the first variable.
This method is particularly useful when one of the equations is easily solvable for one variable. It simplifies the process and reduces potential calculation errors.
linear equations
Linear equations are equations of the first degree, which means they involve variables raised to the power of one. A system of linear equations consists of two or more linear equations with the same set of variables.

For example, consider the system:
  • \(4x - 5y = -11\)
  • \(2x + y = 5\)
Each of these equations represents a straight line when graphically plotted on a coordinate plane.
The solution to the system is the point where the lines intersect, representing the values of the variables that satisfy both equations simultaneously. By using the substitution method, we effectively find this intersection point algebraically, rather than graphically.
solution verification
Once you've found the solution to a system of linear equations, it's crucial to verify that your solution is correct. This involves substituting the values back into the original equations.

Let's look at an example to illustrate how this verification is performed:
  • Our solution gives us \(x = 1\) and \(y = 3\).
  • We substitute \(x = 1\) and \(y = 3\) into the first equation: \(4(1) - 5(3) = -11\).
  • Simplifying, we get \(4 - 15 = -11\), which is true.
  • Next, substitute these values into the second equation: \(2(1) + 3 = 5\).
  • Simplifying, we get \(2 + 3 = 5\), which is also true.
Both equations are satisfied, confirming that our solution \((x, y) = (1, 3)\) is indeed correct. This final step ensures the reliability of our solution.

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Most popular questions from this chapter

Supply and Demand In many applications of economics, as the price of an item goes up, demand for the item goes down and supply of the item goes up. The price where supply and demand are equal is the equilibrium price, and the resulting sup. ply or demand is the equilibrium supply or equilibrium demand. Suppose the supply of a product is related to its price by the equation $$p=\frac{2}{3} q$$ where \(p\) is in dollars and \(q\) is supply in appropriate units. (Here, \(q\) stands for quantity.) Furthermore, suppose demand and price for the same product are related by $$p=-\frac{1}{3} q+18$$ where \(p\) is price and \(q\) is demand. The system formed by these two equations has solution \((18,12),\) as seen in the graph. (GRAPH CANNOT COPY) Find the demand for the electric can opener at each price. (a) \(\$ 6\) (b) \(\$ 11\) (c) \(\$ 16\)

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