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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 11
Chapter 6, Problem 11

Solve each system by substitution.
8x - 10y = -22
3x + y = 6

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1
Start with the given system of equations: \[8x - 10y = -22\] \[3x + y = 6\]
Solve the second equation for \(y\) in terms of \(x\): \[3x + y = 6 \implies y = 6 - 3x\]
Substitute the expression for \(y\) from step 2 into the first equation: \[8x - 10(6 - 3x) = -22\]
Distribute the \(-10\) and simplify the equation: \[8x - 60 + 30x = -22\] Combine like terms to get a single-variable equation in \(x\).
Solve the simplified equation for \(x\), then substitute the value of \(x\) back into the expression for \(y\) from step 2 to find \(y\).

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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 these equations represent lines and their intersections is fundamental to 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 equation is easily solved for one variable.
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Solving Linear Equations

Solving linear equations means isolating the variable to find its value. This involves using inverse operations like addition, subtraction, multiplication, and division. Mastery of these techniques is essential to manipulate equations correctly during substitution.
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Related Practice
Textbook Question

Solve each system by substitution.

7x - y = -10

3y - x = 10

3
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Textbook Question

Graph each inequality. x + 2y ≤ 6

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

Verify that the points of intersection specified on the graph of each nonlinear system are solutions of the system by substituting directly into both equations.

y = 3x2

x2 + y2 = 10

4
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Textbook Question

Find the dimension of each matrix. Identify any square, column, or row matrices.

[24]\(\left\)[ \(\begin{matrix}\) 2 \\ 4 \(\end{matrix}\) \(\right\)]

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 \(\end{matrix}\) \(\right\)]