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Ch. 5 - Systems and Matrices
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 5 - Systems and MatricesProblem 13
Chapter 6, Problem 13

Solve each system by substitution.
7x - y = -10
3y - x = 10

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1
Start with the given system of equations: \(7x - y = -10\) \(3y - x = 10\).
Solve one of the equations for one variable in terms of the other. For example, from the first equation, solve for \(y\): \(7x - y = -10 \implies y = 7x + 10\).
Substitute the expression for \(y\) from step 2 into the second equation: \(3(7x + 10) - x = 10\).
Simplify and solve the resulting equation for \(x\): \(21x + 30 - x = 10\) which simplifies to \(20x + 30 = 10\).
Once you find \(x\), substitute it back into the expression for \(y\) from step 2 to find the value of \(y\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

System of Linear Equations

A system of linear equations consists of two or more linear equations with the same variables. The goal is to find values for the variables that satisfy all equations simultaneously. Understanding how to interpret and manipulate these equations is essential for solving the system.
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Introduction to Systems of Linear Equations

Substitution Method

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This reduces the system to a single equation with one variable, making it easier to solve. It is especially useful when one variable is easily isolated.
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Solving Linear Equations

Solving linear equations means finding the value(s) of the variable(s) that make the equation true. This often involves isolating the variable using inverse operations like addition, subtraction, multiplication, or division. Mastery of these techniques is crucial for solving systems effectively.
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Related Practice
Textbook Question

Graph each inequality. x + 2y ≤ 6

Textbook Question

Find the values of the variables for which each statement is true, if possible.

[−3ab5]=[c04d]\(\left\)[ \(\begin{matrix}\) -3 & a \\ b & 5 \(\end{matrix}\) \(\right\)] = \(\left\)[ \(\begin{matrix}\) c & 0 \\ 4 & d \(\end{matrix}\) \(\right\)]

Textbook Question

Evaluate each determinant.

∣345−2∣\(\left\)| \(\begin{matrix}\) 3 & 4 \\ 5 & -2 \(\end{matrix}\) \(\right\)|

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Textbook Question

Find the dimension of each matrix. Identify any square, column, or row matrices. See the discussion preceding Example 1.

∣−9∣\(\begin{vmatrix}\)-9\(\end{vmatrix}\)

Textbook Question

Solve each system by substitution.

8x - 10y = -22

3x + y = 6

Textbook Question

Are the given matrices inverses of each other? (Hint: Check to see whether their products are the identity matrix In.)

[01000−21−10]and[1011000−10]\(\left\)[ \(\begin{matrix}\) 0 & 1 & 0 \\0 & 0 & -2 \\1 & -1 & 0 \(\end{matrix}\) \(\right\)]\(\quad\) \(\text{and}\) \(\quad\]\left\)[ \(\begin{matrix}\) 1 & 0 & 1 \\1 & 0 & 0 \\0 & -1 & 0 \(\end{matrix}\) \(\right\)]