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Chapter 5: Problem 129

In the following exercises, solve the systems of equations by substitution. $$ \left\\{\begin{array}{l} 3 x-y=-7 \\ 4 x+2 y=-6 \end{array}\right. $$

Short Answer

Expert verified
x = -2, y = 1

Step by step solution

01

Solve one equation for one variable

Take the first equation, which is \(3x - y = -7\). Solve for \(y\) in terms of \(x\):\[ y = 3x + 7 \]
02

Substitute into the second equation

Substitute \(y = 3x + 7\) into the second equation \(4x + 2y = -6\):\[ 4x + 2(3x + 7) = -6 \]
03

Simplify the equation

Simplify the equation from step 2:\[ 4x + 6x + 14 = -6 \]\[ 10x + 14 = -6 \]
04

Solve for \(x\)

Solve the simplified equation for \(x\):\[ 10x = -6 - 14 \]\[ 10x = -20 \]\[ x = -2 \]
05

Substitute \(x\) back into the first equation for \(y\)

Substitute \(x = -2\) into the equation \(y = 3x + 7\):\[ y = 3(-2) + 7 \]\[ y = -6 + 7 \]\[ y = 1 \]
06

Write the solution

Combine the values of \(x\) and \(y\) to write the solution of the system:\( x = -2 \), \( y = 1 \)

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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 used to solve systems of linear equations by substituting one equation into another. This method simplifies the system into a single equation with one variable.

Here鈥檚 a step-by-step approach:
  • First, solve one of the equations for one variable in terms of the other variable. For example, if given the equation \(3x - y = -7\), solve for \(y\) to get \(y = 3x + 7\).
  • Next, substitute this expression into the other equation. In our second equation, \(4x + 2y = -6\), we replace \(y\) with \(3x + 7\), forming \(4x + 2(3x + 7) = -6\).
  • Simplify the resulting equation and solve for the single variable.
  • Lastly, substitute the value obtained back into one of the original equations to solve for the second variable.

The substitution method is particularly useful when one of the equations is easily solved for one of the variables, making it a preferred choice for systems where variables are aligned in a straightforward manner.
linear equations
Linear equations are algebraic expressions that represent straight lines when graphed. They are written in the form \(ax + by = c\) where \(a\), \(b\), and \(c\) are constants.

Key points about linear equations:
  • They involve only first-degree variables, which means no variable is raised to a power other than one.
  • These equations can have one or more variables.
  • The graph of a linear equation in two variables is always a straight line.
  • The solutions to linear equations represent points of intersection of these lines when plotted on a coordinate plane.

In our exercise, we dealt with two linear equations: \(3x - y = -7\) and \(4x + 2y = -6\). By using methods such as substitution, we can find the point where these two lines intersect, representing the solution to the system of equations.
algebraic manipulation
Algebraic manipulation involves the use of algebraic operations to simplify and solve equations. It is fundamental in working through systems of equations in methods like substitution.

Here are some basic algebraic manipulations used in solving our system of equations:
  • Rewriting equations: In step one, we re-arranged \(3x - y = -7\) to solve for \(y\) as \(y = 3x + 7\).
  • Substitution: We replaced \(y\) in the second equation with \(3x + 7\), leading to a single-variable equation.
  • Simplification: In steps two and three, by performing operations like distribution and combining like terms, we reduced the equation \(4x + 2(3x + 7) = -6\) to its simplest form \(10x + 14 = -6\).
  • Solving linear equations: Finally, in steps four and five, we isolated the variable \(x\) and solved, then substituted back to find \(y\).

Mastery of algebraic manipulation is essential for solving more complex mathematical problems and understanding the core structures of algebra.

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Most popular questions from this chapter

Grandpa and Grandma are treating their family to the movies. Matinee tickets cost \(\$ 4\) per child and \(\$ 4\) per adult. Evening tickets cost \(\$ 6\) per child and \(\$ 8\) per adult. They plan on spending no more than \(\$ 80\) on the matinee tickets and no more than \(\$ 100\) on the evening tickets. (a) Write a system of inequalities to model this situation. (b) Graph the system. c) Could they take 9 children and 4 adults to both shows? (a) Could they take 8 children and 5 adults to both shows?

In the following exercises, translate to a system of equations and solve. Brandon has a cup of quarters and dimes with a total value of \(\$ 3.80\). The number of quarters is four less than twice the number of dimes. How many quarters and how many dimes does Brandon have?

Juan is studying for his final exams in Chemistry and Algebra. He knows he only has 24 hours to study, and it will take him at least three times as long to study for Algebra than Chemistry. (a) Write a system of inequalities to model this situation. (b) Graph the system. c) Can he spend 4 hours on Chemistry and 20 hours on Algebra? (d) Can he spend 6 hours on Chemistry and 18 hours on Algebra?

In the following exercises, translate to a system of equations and solve. The difference of two supplementary angles is 24 degrees. Find the measure of the angles.

Laurie was completing the treasurer's report for her son's Boy Scout troop at the end of the school year. She didn't remember how many boys had paid the \(\$ 15\) full-year registration fee and how many had paid the \(\$ 10\) partial-year fee. She knew that the number of boys who paid for a full-year was ten more than the number who paid for a partial-year. If \(\$ 250\) was collected for all the registrations, how many boys had paid the full-year fee and how many had paid the partial-year fee?
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