In this Grade 7 chapter summary from Yoshiwara Intermediate Algebra, students review the core concepts of Chapter 2, including regression lines, interpolation and extrapolation, and methods for solving 2×2 and 3×3 linear systems using substitution, elimination, and Gaussian reduction. Students also consolidate their understanding of graphing linear inequalities using test points and half-planes, as well as solving systems of inequalities by identifying the intersection of shaded regions. The review problems apply these skills to real-world scenarios involving scatterplots, equilibrium price, and constraint-based optimization.
Section 1
📘 Applications of Linear Models
This chapter unites your skills with linear models. You will apply systems of linear equations and inequalities to analyze data, make predictions, and solve complex real-world problems involving multiple variables and constraints.
You'll now work through a set of review problems and interactive examples, challenging you to apply every method from this chapter.
Section 2
Regression line and predictions
We can approximate a linear pattern in a scatterplot using a regression line.
We can use interpolation or extrapolation to make estimates and predictions.
If we extrapolate too far beyond the known data, we may get unreasonable results.
Section 3
Solutions of 2x2 linear systems
A solution to a linear system is an ordered pair that satisfies both equations.
A solution to a linear system is a point where the two graphs intersect.
The substitution method is easier if one of the variables in one of the equations has a coefficient of 1 or -1.
Section 4
Inconsistent and dependent systems
The graphs of the equations in an inconsistent system are parallel lines and hence do not intersect.
The graphs of the two equations in a dependent system are the same line.
If a linear combination of the equations in a system results in an equation of the form
then the system is inconsistent. If an equation of the form
results, then the system is dependent.
Section 5
Graphing linear inequalities
The solutions of the linear inequality
consists of the line and a half-plane on one side of that line.
To Graph an Inequality Using a Test Point.
a. If the resulting statement is true, shade the half-plane that includes the test point.
b. If the resulting statement is false, shade the half-plane that does not include the test point.
Section 6
Solving 3x3 linear systems
A solution to an equation in three variables is an ordered triple of numbers that satisfies the equation.
To solve a linear system, we use linear combinations to reduce the system to triangular form, and then use back-substitution to find the solutions.
Book overview
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Section 1
📘 Applications of Linear Models
This chapter unites your skills with linear models. You will apply systems of linear equations and inequalities to analyze data, make predictions, and solve complex real-world problems involving multiple variables and constraints.
You'll now work through a set of review problems and interactive examples, challenging you to apply every method from this chapter.
Section 2
Regression line and predictions
We can approximate a linear pattern in a scatterplot using a regression line.
We can use interpolation or extrapolation to make estimates and predictions.
If we extrapolate too far beyond the known data, we may get unreasonable results.
Section 3
Solutions of 2x2 linear systems
A solution to a linear system is an ordered pair that satisfies both equations.
A solution to a linear system is a point where the two graphs intersect.
The substitution method is easier if one of the variables in one of the equations has a coefficient of 1 or -1.
Section 4
Inconsistent and dependent systems
The graphs of the equations in an inconsistent system are parallel lines and hence do not intersect.
The graphs of the two equations in a dependent system are the same line.
If a linear combination of the equations in a system results in an equation of the form
then the system is inconsistent. If an equation of the form
results, then the system is dependent.
Section 5
Graphing linear inequalities
The solutions of the linear inequality
consists of the line and a half-plane on one side of that line.
To Graph an Inequality Using a Test Point.
a. If the resulting statement is true, shade the half-plane that includes the test point.
b. If the resulting statement is false, shade the half-plane that does not include the test point.
Section 6
Solving 3x3 linear systems
A solution to an equation in three variables is an ordered triple of numbers that satisfies the equation.
To solve a linear system, we use linear combinations to reduce the system to triangular form, and then use back-substitution to find the solutions.
Book overview
Jump across lessons in the current chapter without opening the full course modal.
Continue this chapter