Systems of Linear Equations and Inequalities

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Questions and Answers

What is the solution to a system of linear equations?

The equation with the highest coefficient

The set of all linear equations

The point of intersection of the lines (correct)

The graph of a single linear equation

What is the purpose of multiplying both equations by necessary multiples in the Elimination Method?

To eliminate one variable (correct)

To solve for both variables simultaneously

To make the coefficients of one variable different

To isolate one variable

What is the first step in the Substitution Method for solving systems of linear equations?

Add or subtract the equations to eliminate one variable

Substitute the expression into the other equation

Solve one equation for one variable (correct)

Graph both equations on the same coordinate plane

What is the purpose of graphing the related equation in the Graphing Method for linear inequalities?

To identify the boundary point Signup and view all the answers

What is the result of multiplying both sides of a linear inequality by a negative number?

The inequality sign is reversed Signup and view all the answers

What is the solution to a linear inequality?

A range of values Signup and view all the answers

What is the value of x in the equation 3(x + 2) - 2 = x - 4(x - 1)?

x = 3 Signup and view all the answers

What is the meaning of the symbol ≥ in an inequality?

Greater than or equal to Signup and view all the answers

What is the geometric term for a point where two rays meet?

Vertex Signup and view all the answers

Which of the following statements is TRUE about angles?

All right angles are congruent. Signup and view all the answers

What is the first step in solving the equation 4x - 7 < 2?

Add 7 to both sides of the equation. Signup and view all the answers

What is the solution to the equation 8x - 41 = 36?

x = 6 Signup and view all the answers

What can be said about the sum of the measures of two acute angles?

It is always greater than 90° Signup and view all the answers

What type of angle has a measure less than 90°?

Acute angle Signup and view all the answers

What can be said about two angles that form a right angle?

They are always complementary Signup and view all the answers

What type of angle has a measure of exactly 90°?

Right angle Signup and view all the answers

What is the result of combining two obtuse angles?

An angle greater than 180° Signup and view all the answers

What is true about angles supplementary to the same angle or congruent angles?

They are always congruent Signup and view all the answers

Study Notes

Solving Systems of Linear Equations

A system of linear equations is a set of two or more linear equations that must be true at the same time.
The solution to a system of linear equations is the point of intersection of the lines, which is the point that satisfies both equations.
Methods for solving systems of linear equations:
- Substitution Method:
  - Solve one equation for one variable.
  - Substitute the expression into the other equation.
  - Solve for the other variable.
- Elimination Method:
  - Multiply both equations by necessary multiples such that the coefficients of one variable are the same.
  - Add or subtract the equations to eliminate one variable.
  - Solve for the other variable.
- Graphing Method:
  - Graph both equations on the same coordinate plane.
  - The point of intersection is the solution.

Solving Linear Inequalities

A linear inequality is an inequality in which the variables are raised to the first power.
The solution to a linear inequality is a range of values that satisfy the inequality.
Methods for solving linear inequalities:
- Graphing Method:
  - Graph the related equation (replace the inequality sign with an equal sign).
  - Identify the boundary point (the point at which the inequality becomes an equality).
  - Shade the region that satisfies the inequality.
- Algebraic Method:
  - Add or subtract the same value to both sides of the inequality.
  - Multiply both sides of the inequality by a positive number.
  - Flip the inequality sign when multiplying both sides by a negative number.
- Interval Notation:
  - Write the solution in interval notation (e.g., (-∞, 3] or [2, ∞)).
  - Use parentheses to indicate that the endpoint is not included.
  - Use brackets to indicate that the endpoint is included.

Key Concepts

The inequality sign can be reversed when multiplying both sides of the inequality by a negative number.
The solution to a system of linear equations can be a single point, no solution, or infinitely many solutions.
The solution to a linear inequality can be expressed in interval notation, graphically, or in set-builder notation.

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Description

Learn how to solve systems of linear equations using the substitution, elimination, and graphing methods. Also, discover how to solve linear inequalities using graphing and algebraic methods, and understand key concepts related to these topics.