Graphing Systems of Equations PDF

Here is the transcription of the provided text, formatted in Markdown: # Lesson 7-1 ## Graphing Systems of Equations ### Explore Intersections of Graphs **Online Activity:** Use graphing technology to complete the Explore. **INQUIRY:** How can you solve a linear equation by graphing? ### Learn G...

Here is the transcription of the provided text, formatted in Markdown: # Lesson 7-1 ## Graphing Systems of Equations ### Explore Intersections of Graphs **Online Activity:** Use graphing technology to complete the Explore. **INQUIRY:** How can you solve a linear equation by graphing? ### Learn Graphs of Systems of Equations A set of two or more equations with the same variables is called a **system of equations**. An ordered pair that is a solution of both equations is a solution of the system. A system of two linear equations can have one solution, an infinite number of solutions, or no solution. * A system of equations is **consistent** if it has at least one ordered pair that satisfies both equations. *One solution* * If a consistent system of equations has **exactly one solution**, it is said to be **independent**. The graphs intersect at one point. * If a consistent system of equations has an infinite number of solutions, it is **dependent**. The graphs are the same line. This means that there are unlimited solutions that satisfy both equations. * A system of equations is **inconsistent** if it has no ordered pair that satisfies both equations. The graphs are parallel. *It has no solution.* ### Example 1 Consistent Systems Use the graph to determine the number of solutions the system has. Then state whether the system of equations is consistent or inconsistent and if it is independent or dependent. $y = -3x + 1 \\ y = x - 3$ The graph shows two lines intersecting at one point. One solution. Therefore, the system is **consistent and independent.** --- **Today's Goals** * Determine the number of solutions of a system of linear equations. * Solve systems of equations by graphing. * Solve linear equations by graphing systems of equations. * Use graphing calculators to solve systems of equations. **Today's Vocabulary** * system of equations * consistent * independent * dependent * inconsistent --- ### Example 2 Inconsistent Systems Use the graph to determine the number of solutions the system has. Then state whether the system of equations is consistent or inconsistent and if it is independent or dependent. $y = 1/2x + 2 \\ y = 1/2x - 1$ Since the graphs of these two lines are parallel, there is **no solution** of the system. Therefore, the system is **inconsistent** (*No Solution*). #### Check Determine whether each graph shows a system that is consistent or inconsistent and if it is independent or dependent. * A graph shows two parallel lines which have are described as: *Same slope, different y-intercept, Parallel, no solution, inconsistent.* * A graph shows two intersecting lines which are described as: *Different slope, different y-intercept, one solution, consistent, independent.* Described by the equations: $y=-x-1,y=x-4$ ### Example 3 Number of Solutions, Equations in Slope-Intercept Form $y = mx + b$ Determine the number of solutions the system has. Then state whether the system of equations is consistent or inconsistent and if it is independent or dependent. $y = 6x + 10 \\ y = 6x + 4$ Because **the slopes are the same** and **the y-intercepts are different**, the lines are **parallel.** The system has **no solution**. Therefore, the system is inconsistent. #### Check Determine the number of solutions the system has. $y = 5/4x + 1/4$ *One Solution, different slope, different y-intercept,* thus a *consistent and independent* system ### Example 4 Number of Solutions, Equations in Standard Form Determine the number of solutions the system has. Then state whether the system of equations is consistent or inconsistent and if it is independent or dependent. $4y - 6x = 16 \\ 9x - 6y = -24$ Write both equations in slope-intercept form. $4y - 6x = 16$ Original equation $9x - 6y = -24$ $4y = 6x + 16$ Simplify $-6y = -9x - 24$ $y = 3/2x + 4$ Simplify $y = 3/2 + 4$ Because the slopes are the same and the y-intercepts are the same, this is the same line. Since the graphs of these two lines are the same, there are infinitely many solutions. Therefore, the system is **consistent and dependent.** #### Check Determine the number of solutions each system has. $4x - 8y = 16 \\ 6x - 12y = 5$ The solution for the above system is **No Solution**. ### Learn Solving Systems of Equations by Graphing You can solve a system of equations by graphing each equation carefully on the same coordinate plane. Every point that, lies on the line of one equation represents a solution of that equation. Similarly, every point on the line of the second equation in a system represents a solution of that equation. Therefore, the solution of a system of equations is the point at which the graphs intersect. For example, the solution of this system is (-1, 3). That is the point at which the graphs intersect. Since the point of intersection lies on both lines, the ordered pair satisfies each equation in the system. $y = -2x + 1, y = x + 4$ --- ### Example 5 Solve a System by Graphing Graph the system and determine the number of solutions that it has. If it has one solution, determine its coordinates. $y = -2x + 14 \\ y = 3/2x + 1$ The slope for the first equation is -2 (down) and the y-intercept is (0, 14). The slope for the second equation is 3/2 and the y-intercept is (0, 1). The graphs of the lines appear to intersect at the point (5, 4). If you substitute 5 for x and 4 for y into the equations, both are true. Therefore, (5, 4) is the solution of the system. **Check** Graph the system of equations for: 1. $3x + 5y = 10$ 2. $x - 5y = -10$ This translates to the equations: $y=(-3/5)x+2$ and $y=(1/5)x+2$, which gives us a slope of -3/5 down and 1/5 right. It also shows us that their y-intercept is (0, 2). This is a *Consistent system*. ### Example 6 Graph and Solve a System of Equations Graph the system and determine the number of solutions that it has. If it has one solution, determine its coordinates. $-3x + 2y = 12 \\ 6x - 4y = 8$ The graphed equations are: $y=(3/2)x+6 , y=(3/2)x-2$ which yield equations that have the same slopes and different y-intercepts meaning the graphs have *No Solution*. The lines have the same slope but different y-intercepts, so the lines are parallel. Since they do not intersect, this system has **no solution** *inconsistent system*. --- **Def** Here is a summary of graphing systems of equations: * **Consistent:** * One Solution (Independent): Different Slope and Different y-intercept * An infinite number of solutions (dependent): Same Slope and Same y-intercept * **Inconsistent:** No Solution: *Same Slope and Different y-intercept* --- # Lesson 7.2 ### Substitution Explore Using Substitution **Online Activity:** Use a system of equations to complete the Explore. **Inquiry:** How can you rewrite a system of equation as a single equation with only one variable? ## Learning: Solving System of Equations by Substitution Exact solutions result when algebraic methods are used to solve systems of equations. One algebraic method is called substitution. ### Key concept: Method of substitution 1. Where necessary, solve at least one equation for one variable; 2. Substitute The resulting expression from Step 1 into the other equations to replace the variable. Then solve the equation, 3. Substitute the value from Step 2 into either equation, and solve for the other variable. Write the solution as an ordered pair. ### Substitution with Equation Example 1 Use substitution to solve the systems of equations: $3x - y = -7; y = 4x + 11$ **Step 1:** The second equation is already solved for y. **Step 2:** Substitute $4x + 11$ for $y$ in the first equation. $3x - (4x + 11) = -7$ $3x - 4x - 11 = -7$ $-x - 11 = -7$ $x = -4$ Multiply each side by -1 **Step 3:** Substitute $-4$ for $y$ in either equation to find $y$. $y = 4(-4) + 11 = -16 + 11 = -5$ The solution is $(-4, -5)$ --- **Today's Goals** * Solve systems of equations by using the subsitution method. **Today's Vocabulary** * Substitution --- ### Example 2: Solve and Then Substitute Use substitution to solve system of equations $5x -3y = -25 \\ x + 4y = 18$ **Step 1:** Solve the second equation for x since the coefficient is 1. $x + 4y = 18 \\ x = 18 - 4y$ **Step 2:** Substitute $18 - 4y$ for $x$ in the equation. $5x - 3y = -25 => 5(18 - 4y) - 3y = -25$ simplify: $90 - 20y - 3y = -25 -> -23y = -115 => y = 5$ **Step 3:** Substitute 5 for y in either equation to find $x$. This can be done by using the equation: $x + 4y = 18 => x + 4(5) = 18 => x = -2$ The solution is thus the coordinate $(-2, 5)$. **Check** Use subsitution to solve the following system of equations: $5x + 3y = 5 \\ x + 2y = -13$ In order to check the math behind the subsitution to solve the equations is accurate. ### Example 3 Use Substitution When There are No or Many Solutions Use substitution to solve the system of equations. $4x + 2y = -8 \\ y = -2x -4$ Substitute (-2x-4) for y in the first equation. $4x + 2(-2x -4) = -8 => 4x - 4x -8 = -8 => -8 = -8$ The equation $\text{-8 = -8}$ is an identity. Thus, there are an infinite number of solutions. ### Check Select the correct statement about the system of equations. $-x + 2y = 2 => y = \frac{1}{2}x + 1 $ This system has infinitely many solutions. --- # Lesson 7-3 Elimination Using Addition and Subtraction # Learn Solving Systems of Equations by Elimination with Addition ## The elimination method involves eliminating a variable by combining the individual equations within a system of equations. One way to combine equations is by using addition. ### Key Concept Elimination Method Using Addition * Write the system so like terms with opposite coefficients are aligned. * Add the equations, eliminating one variable. Then solve the equation. * Substitute the value from step 2 into one of the equations and solve for the other variable. Write the solution as an ordered pair. ### elimination example #1 Using Addition Use elimination to solve the system of equations. $3x + 5y = 11 \\ 5x - 5y = 5$ Align terms with opposite coefficients. Since 5y and -5y have opposite coefficients. Add the equations. $3x + 5y = 11 \\ (+) 5x - 5y = 5$ 8x = 16 x = 2 Solve for the other variable $3x + 5y = 11 \\ 3(2) + 5y = 11$ $6 + 5y = 11$ $5y = 5$ $y = 1$ The solution is (2, 1) --- ### Today's Goals * Solve systems of equations by eliminating a variable using addition * Solve systems of equations by eliminating a variable using subtraction **Today's Vocabulary** *Elimination* --- ### Learn Solving Systems of Equations by Elimination With Subtraction. When the coefficiants on a variable are the same in two equations, you can eliminate the variable by subtracting one equations from the other Key concept. Elimination Method Using Subtraction 1. Write the system so like terms with the same coefficiants are aligned. 2. Subtract one equation from the other eliminating one variable. Then solve the equation. 3. Subsitute the value from Step 2 into one of the equations and solve for the other variable. Write the solution as an ordered pair. ### Elimination Using Subtraction Example Use elimination to solve the system of equation $3x + 6y = 30 \\ 5x + 6y = 6$ Align terms with the same coefficients. Since 69 and 60 have the same coefficients you can $subtract$ the equation to eliminate variable Subtract the equations from one another $$3x + 6y = 30 =>\newline (-)\newline 5x + 6y = 6 $$$ Divide each side by -2 Simplify $x=-12$ Substitute -12 for x either equation to find the value of 4 $3x + 6y = 30 => 3(-12) + 6v = 30" $3(-12) + 6y = 30 =>3(-12) + 6y = 30$ Multiply Add 36 to each side simplify. $ y = 11$. The solution is (-12,11). Use elimination to solve the system of equations # Lesson 7-4: Elimination Using Multiplication ### Explore Graphing and Elimination Using Multiplication **Online Activity:** Use an interactive tool to complete the Explore. **INQUIRY** How can you produce a new system of equations with the same solution as the given system? ### Learn Solving Systems of Equations by Elimination with Multiplication **Key Concept:** Elimination Method Using Multiplication **Step 1:** Multiply at least one equation by a constant to get two equations that contain opposite terms. **Step 2:** Add the equations, eliminating one variable. Then solve the equations. **Step 3:** Substitute the value from Step 2 into one of the equations and solve for the other variable. Write the solution as an ordered pair. **Today's Goal** *Solve systems of equations by eliminating a variable using multiplication and addition.* ### Example 1 Elimination Using Multiplication Use elimination to solve the system of equations: $10x + 5y = 30 \\ 5x - 3y = -7$ STEP 1: Multiply an equation by a constant: The coefficients of x will be opposites if the second equation is multiplied by -2 $5x - 3y = -7 => (-2)(5x-3y) = (-2)(-7) => -10x+6y=14$ STEP 2: Add the equations: $10x + 5y = 30 => (+)-10x+6y=14 => 11y = 44 => y = 4$ ### Example 2 Multiply Both Equations to Eliminate a Variable Use elimination to solve the system of equations: $3x + 4y = -22 \\ -2x + 3y = -8$ ### Step 1 Multiply both squations by a constant. $3x+4y = -22, Multiply by a Constant, -2x+3y = -8, Distributive Property, Simplify$ ### Step 2 Add the equations $6x-8y = -44 \\ (+)-6x+9y = -24$ The variable $x$ is eliminated. Distribute each side by $17, y = -4 $ --- ### Lesson 7-5: System of Inequalities **Explore Solutions Of System of Inequalities** Online activity use an interactive tool to complete the Explore Inquiry: HOw are the solutions of the system of inequalities represented by the graph ### Learn Solving Systems Of Inequalities by Graphing Set of two or more inequalities with the same variables is a system of inequalities The solution of the system of inequalities with two variable is the set of order pairs that satisfy the inequalities of a system The solution is represent by the overlap or intersection or the graph of the inequalities Graph one inequality of of the system The boundary by $x - 2y > -6$ is dashed and in not included if graph Today goal; Solve system if lines Example; Solving a systems of inequAlities is graphing - 2x+1 intersects at (3,4) is 13 included in the solution. Is a boundary (solid) graph one quality of your system? 23 graphs. He is not an included because the **The region and is not included *Graph equation from the system * The bound is y equal to S.I is does and is not included in the solution. So you see in point (0,0) which -Is equal to Y and y is right and x and any points If any . 3/y5 and y+5 ### Check, graph of the systems of inequalities Text points (0,1) and (2,3) ### Solution for system Inequalities (0, 0).