Systems of Linear Equations
Chapter 7 Algebra 1, Resources - Systems of Equations and Inequalities
TEKS Focus
In Algebra 1, students analyze situations and formulate systems of linear equations in two unknowns to solve problems. A.8.A. TAKS Analysis again reveals that students struggle with this expectation.
In Lessons 7-1 through 7-3 students solve problems involving systems of linear equations using graphing, substitution and elimination of one variable. In Lesson 7-4, students apply the appropriate method to solve real world problems. Place special emphasis on these lessons because they focus on A.8.A.
New for Algebra 1, students investigate methods for solving linear inequalities using graphs A.7.B, Student us graphs to solve systems of linear inequalities in Lesson 7-5. a.7.B
Big Ideas Overview for Systems of Equations and Inequalities
This chapter examines systems of two linear equations and system of two linear inequalities. Students first graph two equations on a single coordinate system and identify the point of intersection to find the solution. When the solution to the system is not apparent graphically, algebraic methods such as substitution and elimination enable students to find the coordinates of the point of intersection. Students will learn that some systems may be solved easily by graphing, while other systems may need to be solved algebraically or with a graphing calculator.
7.1 Solving Systems by Graphing A.8.A. A.8.B, A.8.C Objective 4
Two or more linear equations together (with the same variables) are called a system of linear equations. The solution to a system of linear equations is any ordered pair that satisfies all equations in the system.
The graph of a system of linear equations can take three distinct forms: the lines intersect so there is one solution. The lines are parallel so there are no solutions, or their lines are the same so there are infinitely many solutions.
Students often find an infinite number of solutions perplexing. Remind them that the solution to a linear equation such as y = 2x + 1 is the infinite set of ordered pairs (x, y). that makes the equation true. The graph of this solution set forms a line that never ends. So two lines that lie on top of one another intersect at each and every point.
Consider reviewing with students the y-intercept form of a linear equation, y = mx + b, and how to graph a line. For example, find the slope m and y-intercept b, plot the point (0.b) use the slope as a ratio of rise over run to plot one or more points, and connect the points with a straight line. Students can also use the slopes and y-intercepts to identify the number of solutions in a system: different slopes mean the lines intersect at one point, same slope but different y-intercepts mean the lines are parallel and have no intersections, and same slope and same y-intercept means the lines have infinitely many solutions.
7.2 Solving Systems Using Substitution
A.8.A, A.8.B, Objective 4
The substitution method involves replacing one variable with an equivalent expression containing only the other variable, solving for the remaining variable, and then substituted the value for the other variable into either original equation, and finally solving for the value of the original variable.
Some students have difficulty deciding which equation and variable to being solving for. Have students identify al l variables with coefficients of 1 0r -1 and choose one of these to solve for...
You may also want to present that systems of equations with no solution result in false equations, such as 7 = -2, while systems with infinitely many solutions will result in a universally true equation, such as 3 + 2 = 5. In both special cases, the variable is eliminated from the equation.
7.3 Solving Systems Using Elimination TEKS A.8(A). A.8.B, A.8.C Obj. 4
As with the substitution method, the goal of the elimination method is to obtain a one-variable equation, to solve for the value of the variable, and to substitute this value into either original equation to find the value of the other variable. The primary difference between the methods involves how the one-variable equation is obtained.
Guide students in identifying which variable is easiest to eliminate. When coefficients are not the same or not opposites, multiplication is needed. Remind students to multiply all terms on both sides of the equation, since they are applying the Multiplication Property of Equality.
Many students also find subtraction of polynomials difficult. Remind students that subtracting a negative is the same as adding a positive. (-- ) Consider showing students how they can take the opposite of the entire polynomial and add. Point out that this is simply another application of the Multiplication Property of Equality using -1 as the multiplier.
7.4 Application of Linear Systems
A.8.A, A.8.B, A.8.C. Obj 4.
In theory, any system of linear equations can be solved by graphing and then reading the coordinates of the point of intersection form the graph. However, unless the x-value and y-values of the solution are integers, accurately determining the value of the coordinates may be difficult. The TRACE function of a graphing calculator can be used for an approximate solution.
Similarly, any system of linear equations can be solved by substitution, but the calculations may be quite difficult and lengthy when neither equation has a variable with a coefficient of 1 or -1, such as rewriting 7x + 3y = 13 -3/7y + 13/7.
As with the other methods, it is theoretically possible to solve any system of two linear equations by first multiplying to make the coefficients of one of the variables into additive inverses, then using the elimination method. However, the calculations again may be too cumbersome to make this the best approach.
Finally, there are some systems that can be reasonably solved by any one of the methods taught. One student might decide to use substitution while another might elect to multiply an equation and use elimination.
7.5 Linear Inequalities
TEKS a.1.C, a.1.D, Obj 1, A.7.B Obj. 4
Many students benefit from a comparison of linear inequalities in one variable with linear inequalities in two variables, as shown initially in these lessons. Emphasize that One-variable solutions are graphed on a coordinate plane.
A linear inequality in two variables is different from a linear equation in two essential ways. First, the solution set is a region defined by a boundary line rather than a point or line. Second, multiplying both sides of an inequality by a number less than zero reverses the direction of the inequality symbol.
In general, when an inequality is written so that y is isolated on the left, the < and < symbols mean that the solution region is below the boundary line and the > and > symbols mean the solution is above the boundary line. Use a dashed line for < and >, and a solid line for < and > symbols. The < and > symbols indicate that the boundary line is included in the solution region while < and > symbols indicate the boundary line is not included.
7.6 Systems of Linear Inequalities
Two or more linear inequalities together are called a system of linear inequalities. When picturing the solution region for a system of linear inequalities, it is visually powerful to use different colors to help clarify exactly which points make up the solution set. Tit is also important to emphasize whether or not points on the various boundary lines of the regions are included in the solution set by using either solid or dashed lines.
Because it easy for students to forget to reverse the inequality sign when multiplying a negative, it is particularly important to test points in the solutions region. It is also possible to graph a set of linear inequalities on the graphing calculator, using the shaded area to confirm test points.
TEKS Focus
In Algebra 1, students analyze situations and formulate systems of linear equations in two unknowns to solve problems. A.8.A. TAKS Analysis again reveals that students struggle with this expectation.
In Lessons 7-1 through 7-3 students solve problems involving systems of linear equations using graphing, substitution and elimination of one variable. In Lesson 7-4, students apply the appropriate method to solve real world problems. Place special emphasis on these lessons because they focus on A.8.A.
New for Algebra 1, students investigate methods for solving linear inequalities using graphs A.7.B, Student us graphs to solve systems of linear inequalities in Lesson 7-5. a.7.B
Big Ideas Overview for Systems of Equations and Inequalities
This chapter examines systems of two linear equations and system of two linear inequalities. Students first graph two equations on a single coordinate system and identify the point of intersection to find the solution. When the solution to the system is not apparent graphically, algebraic methods such as substitution and elimination enable students to find the coordinates of the point of intersection. Students will learn that some systems may be solved easily by graphing, while other systems may need to be solved algebraically or with a graphing calculator.
7.1 Solving Systems by Graphing A.8.A. A.8.B, A.8.C Objective 4
Two or more linear equations together (with the same variables) are called a system of linear equations. The solution to a system of linear equations is any ordered pair that satisfies all equations in the system.
The graph of a system of linear equations can take three distinct forms: the lines intersect so there is one solution. The lines are parallel so there are no solutions, or their lines are the same so there are infinitely many solutions.
Students often find an infinite number of solutions perplexing. Remind them that the solution to a linear equation such as y = 2x + 1 is the infinite set of ordered pairs (x, y). that makes the equation true. The graph of this solution set forms a line that never ends. So two lines that lie on top of one another intersect at each and every point.
Consider reviewing with students the y-intercept form of a linear equation, y = mx + b, and how to graph a line. For example, find the slope m and y-intercept b, plot the point (0.b) use the slope as a ratio of rise over run to plot one or more points, and connect the points with a straight line. Students can also use the slopes and y-intercepts to identify the number of solutions in a system: different slopes mean the lines intersect at one point, same slope but different y-intercepts mean the lines are parallel and have no intersections, and same slope and same y-intercept means the lines have infinitely many solutions.
7.2 Solving Systems Using Substitution
A.8.A, A.8.B, Objective 4
The substitution method involves replacing one variable with an equivalent expression containing only the other variable, solving for the remaining variable, and then substituted the value for the other variable into either original equation, and finally solving for the value of the original variable.
Some students have difficulty deciding which equation and variable to being solving for. Have students identify al l variables with coefficients of 1 0r -1 and choose one of these to solve for...
You may also want to present that systems of equations with no solution result in false equations, such as 7 = -2, while systems with infinitely many solutions will result in a universally true equation, such as 3 + 2 = 5. In both special cases, the variable is eliminated from the equation.
7.3 Solving Systems Using Elimination TEKS A.8(A). A.8.B, A.8.C Obj. 4
As with the substitution method, the goal of the elimination method is to obtain a one-variable equation, to solve for the value of the variable, and to substitute this value into either original equation to find the value of the other variable. The primary difference between the methods involves how the one-variable equation is obtained.
Guide students in identifying which variable is easiest to eliminate. When coefficients are not the same or not opposites, multiplication is needed. Remind students to multiply all terms on both sides of the equation, since they are applying the Multiplication Property of Equality.
Many students also find subtraction of polynomials difficult. Remind students that subtracting a negative is the same as adding a positive. (-- ) Consider showing students how they can take the opposite of the entire polynomial and add. Point out that this is simply another application of the Multiplication Property of Equality using -1 as the multiplier.
7.4 Application of Linear Systems
A.8.A, A.8.B, A.8.C. Obj 4.
In theory, any system of linear equations can be solved by graphing and then reading the coordinates of the point of intersection form the graph. However, unless the x-value and y-values of the solution are integers, accurately determining the value of the coordinates may be difficult. The TRACE function of a graphing calculator can be used for an approximate solution.
Similarly, any system of linear equations can be solved by substitution, but the calculations may be quite difficult and lengthy when neither equation has a variable with a coefficient of 1 or -1, such as rewriting 7x + 3y = 13 -3/7y + 13/7.
As with the other methods, it is theoretically possible to solve any system of two linear equations by first multiplying to make the coefficients of one of the variables into additive inverses, then using the elimination method. However, the calculations again may be too cumbersome to make this the best approach.
Finally, there are some systems that can be reasonably solved by any one of the methods taught. One student might decide to use substitution while another might elect to multiply an equation and use elimination.
7.5 Linear Inequalities
TEKS a.1.C, a.1.D, Obj 1, A.7.B Obj. 4
Many students benefit from a comparison of linear inequalities in one variable with linear inequalities in two variables, as shown initially in these lessons. Emphasize that One-variable solutions are graphed on a coordinate plane.
A linear inequality in two variables is different from a linear equation in two essential ways. First, the solution set is a region defined by a boundary line rather than a point or line. Second, multiplying both sides of an inequality by a number less than zero reverses the direction of the inequality symbol.
In general, when an inequality is written so that y is isolated on the left, the < and < symbols mean that the solution region is below the boundary line and the > and > symbols mean the solution is above the boundary line. Use a dashed line for < and >, and a solid line for < and > symbols. The < and > symbols indicate that the boundary line is included in the solution region while < and > symbols indicate the boundary line is not included.
7.6 Systems of Linear Inequalities
Two or more linear inequalities together are called a system of linear inequalities. When picturing the solution region for a system of linear inequalities, it is visually powerful to use different colors to help clarify exactly which points make up the solution set. Tit is also important to emphasize whether or not points on the various boundary lines of the regions are included in the solution set by using either solid or dashed lines.
Because it easy for students to forget to reverse the inequality sign when multiplying a negative, it is particularly important to test points in the solutions region. It is also possible to graph a set of linear inequalities on the graphing calculator, using the shaded area to confirm test points.
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