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Ch. 5 - Systems of Equations and Inequalities
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 5 - Systems of Equations and InequalitiesProblem 17
Chapter 6, Problem 17

In Exercises 5–18, solve each system by the substitution method. y = (1/3)x + 2/3 y = (5/7)x - 2
Two equations for a system of linear equations: y = (1/3)x + 2/3 and y = (5/7)x - 2.

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Step 1: Since both equations are equal to y, set the right-hand sides of the equations equal to each other: 13x + 23 = 57x - 2.
Step 2: To solve for x, first eliminate the fractions by finding a common denominator or multiply both sides of the equation by the least common multiple of the denominators (which is 21) to clear the fractions.
Step 3: After clearing the fractions, simplify the resulting equation and isolate the variable x on one side.
Step 4: Solve the simplified linear equation to find the value of x.
Step 5: Substitute the value of x back into either original equation (for example, y = 13x + 23) to find the corresponding 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 solution is the set of values that satisfy all equations simultaneously. Graphically, this corresponds to the point(s) where the lines intersect.
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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.
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Slope-Intercept Form

Equations in slope-intercept form are written as y = mx + b, where m is the slope and b is the y-intercept. Understanding this form helps in interpreting the equations and solving systems by substitution or graphing.
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Related Practice
Textbook Question

Solve each system in Exercises 5–18. {3(2x+y)+5z=−12(x−3y+4z)=−94(1+x)=−3(z−3y)\(\begin{cases}\)3(2x + y) + 5z = -1 \\2(x - 3y + 4z) = -9 \\4(1 + x) = -3(z - 3y)\(\end{cases}\)

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Textbook Question

Write the partial fraction decomposition of each rational expression. x/(x2 +2x -3)

Textbook Question

Write the partial fraction decomposition of each rational expression. 4x2+13x-9/x (x − 1)(x+3)

Textbook Question

Graph each inequality. (x−2)2+(y+1)2<9

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Textbook Question

In Exercises 16–24, write the partial fraction decomposition of each rational expression. (4x^2 - 3x - 4)/x(x + 2)(x - 1)

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Textbook Question

In Exercises 1–18, solve each system by the substitution method. {x+y=1(x−1)2+(y+2)2=10\(\begin{cases}\)x + y = 1 \\(x - 1)^2 + (y + 2)^2 = 10\(\end{cases}\)