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Chapter 4: Problem 69

Solve each system by the substitution method. $$\begin{array}{l}3 x+y=2 \\\\-x-3 y=6\end{array}$$

Short Answer

Expert verified
The solution is \( \left( \frac{3}{2}, -\frac{5}{2} \right) \).

Step by step solution

01

Solve One Equation for One Variable

First, choose one of the equations to solve for one variable. Let's solve the first equation for y: \[ 3x + y = 2 \] Rewrite it as: \[ y = 2 - 3x \]
02

Substitute into the Other Equation

Now that y is expressed in terms of x, substitute \( y = 2 - 3x \) into the second equation: \[ -x - 3(2 - 3x) = 6 \]
03

Simplify and Solve for x

Expand and simplify the equation: \[ -x - 6 + 9x = 6 \] Combine like terms: \[ 8x - 6 = 6 \] Solve for x: \[ 8x = 12 \] \[ x = \frac{12}{8} = \frac{3}{2} \]
04

Substitute Back to Find y

Substitute \( x = \frac{3}{2} \) back into the expression for y: \[ y = 2 - 3 \left(\frac{3}{2}\right) \] Simplify to get y: \[ y = 2 - \frac{9}{2} \] \[ y = \frac{4}{2} - \frac{9}{2} = -\frac{5}{2} \]
05

Write the Solution as an Ordered Pair

The solution to the system of equations is: \( x = \frac{3}{2} \) and \( y = -\frac{5}{2} \). Thus, the solution is \( \left( \frac{3}{2}, -\frac{5}{2} \right) \).

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Key Concepts

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

system of equations
A system of equations is a set of two or more equations with the same variables. The goal is to find values that satisfy all equations simultaneously. In this exercise, we have a system of two linear equations with variables x and y:
  • 3x + y = 2
  • -x - 3y = 6
Systems of equations can be solved using different methods, such as substitution, elimination, or graphing. Here, we use the substitution method. By solving these equations, we find the exact values of x and y that make both equations true.
algebraic manipulation
Algebraic manipulation involves rearranging equations and expressions using algebraic rules. It helps simplify and solve equations step by step. In this exercise, algebraic manipulation is used extensively:
  • First, we solve the first equation for y: 3x + y = 2 becomes y = 2 - 3x.
  • Next, we substitute this value into the second equation: -x - 3(2 - 3x) = 6.
  • We then expand, simplify, and solve for x: -x - 6 + 9x = 6 simplifies to 8x - 6 = 6, and then 8x = 12, hence x = 3/2.
By performing these steps, we use algebraic rules to isolate and solve for each variable.
solving linear equations
Solving linear equations means finding the values of variables that satisfy a linear equation. In the substitution method, we solve by substituting one variable from one equation into another equation.
  • First, express one variable in terms of the other: y = 2 - 3x.
  • Substitute this expression into the second equation: -x - 3(2 - 3x) = 6.
  • Expand and simplify: -x - 6 + 9x = 6 simplifies to 8x - 6 = 6. Solve for x: 8x = 12, so x = 3/2.
  • Finally, substitute the x value back into the expression for y: y = 2 - 3(3/2) = 2 - 9/2 = -5/2.
By carefully following each step, we find the consistent solution for both equations.
ordered pairs
An ordered pair (x, y) represents a point in a two-dimensional coordinate system where x is the horizontal position and y is the vertical position. The final solution to the system of equations is given as an ordered pair.
  • Our solved values for x and y are x = 3/2 and y = -5/2.
  • Thus, the solution to the system is (3/2, -5/2).
This ordered pair is where the graphs of the two linear equations intersect, showing the point that satisfies both equations. Writing the solution as an ordered pair clearly communicates the answer in the context of a coordinate system.

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Most popular questions from this chapter

Weighing dogs. Cassandra wants to determine the weights of her two dogs, Mimi and Mitzi. However, neither dog will sit on the scale by herself. Cassandra, Mimi, and Mitzi altogether weigh 175 pounds. Cassandra and Mimi together weigh 143 pounds. Cassandra and Mitzi together weigh 139 pounds. How much does each weigh individually?

Solve each system using the Gauss-Jordan elimination method. $$ \begin{array}{c} x-y=-1 \\ 2 x-y=2 \end{array} $$

Discussion. Which of the following equations is inconsistent with the equation \(3 x+4 y=8 ?\) a) \(y=\frac{3}{4} x+2\) b) \(6 x+8 y=16\) c) \(y=-\frac{3}{4} x+8\) d) \(3 x-4 y=8\)

Use Cramer's rule to solve each system. \(\begin{aligned} x+y+z &=0 \\ 2 y+2 z &=0 \\ 3 x-y &=-1 \end{aligned}\)

Write the system of equations represented by each augmented matrix. $$ \left[\begin{array}{rr|r} 5 & 1 & -1 \\ 2 & -3 & 0 \end{array}\right] $$
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