Math Problem Statement
Practice Problems (Solving Systems of Equations)
- Identty a solution to this system ofequations: {x+3= 73 エーアニ-7 a. (-5,2) b. (-2,5) c. (-3,4) d. (4,-3)
- Lin is solving this system of equations: (3x - 2y =3% She starts by rearranging the second equation to isolate the y variable: y = 4 - 1.5x. She then substituted the expression 4 - 1.5x for y in the first equation, as shown: 6x - 5(4 - 1.5x) 6x - 20 - 7.5x -1.5x = 34 = 34 = 54 = -36 У У = 4 - 1.5x = 4ー1.5・（-36） = 58 а. Check to see if Lin's solution of (-36,58) makes both equations in the system true. b. If your answer to the previous question is "no," find and explain her mistake. If your answer is "yes," graph the equations to verify the solution of the system.
- Solve each system of equations. (2x - 4y = 20 a. x = 4 y = 6x + 11 b. (2x - 3y = 7
Solution
Let's solve these system of equations problems step by step.
Problem 1: Identify a solution to the system of equations
The given system of equations is:
x + 3 &= 7 \\ y - x &= -7 \end{aligned}$$ Let's solve this step by step. #### Step 1: Solve the first equation From the first equation: $$x + 3 = 7$$ Subtract 3 from both sides: $$x = 7 - 3 = 4$$ #### Step 2: Solve for $$y$$ Now, substitute $$x = 4$$ into the second equation: $$y - x = -7$$ $$y - 4 = -7$$ Add 4 to both sides: $$y = -7 + 4 = -3$$ So the solution is $$(x, y) = (4, -3)$$. The correct answer is: **d. (4, -3)** --- ### Problem 2: Check Lin's solution and find any mistakes The system of equations is: $$\begin{aligned} 6x - 5y &= 34 \\ y &= 4 - 1.5x \end{aligned}$$ Lin's solution is $$(-36, 58)$$. Let's check if this makes both equations true. #### Step 1: Check the second equation We are given that $$y = 4 - 1.5x$$. Substitute $$x = -36$$: $$y = 4 - 1.5(-36)$$ $$y = 4 + 54 = 58$$ So, Lin's value of $$y = 58$$ is correct according to the second equation. #### Step 2: Check the first equation Now, substitute $$x = -36$$ and $$y = 58$$ into the first equation $$6x - 5y = 34$$: $$6(-36) - 5(58) = 34$$ $$-216 - 290 = -506$$ This does **not** equal 34. So, Lin's solution is **incorrect**. #### Step 3: Find Lin's mistake The mistake occurred when Lin substituted $$y = 4 - 1.5x$$ into the first equation. The equation after substitution became: $$6x - 5(4 - 1.5x) = 34$$ Expanding: $$6x - 20 + 7.5x = 34$$ Combine like terms: $$13.5x - 20 = 34$$ Add 20 to both sides: $$13.5x = 54$$ Now divide by 13.5: $$x = \frac{54}{13.5} = 4$$ Substitute $$x = 4$$ into $$y = 4 - 1.5x$$: $$y = 4 - 1.5(4) = 4 - 6 = -2$$ So the correct solution is $$(4, -2)$$. --- ### Problem 3: Solve each system of equations #### (a) System: $$\begin{aligned} 2x - 4y &= 20 \\ x &= 4 \end{aligned}$$ Since $$x = 4$$, substitute $$x = 4$$ into the first equation: $$2(4) - 4y = 20$$ $$8 - 4y = 20$$ Subtract 8 from both sides: $$-4y = 12$$ Divide by -4: $$y = \frac{12}{-4} = -3$$ So, the solution is $$(4, -3)$$. #### (b) System: $$\begin{aligned} 2x - 3y &= 7 \\ y &= 6x + 11 \end{aligned}$$ Substitute $$y = 6x + 11$$ into the first equation: $$2x - 3(6x + 11) = 7$$ Distribute the -3: $$2x - 18x - 33 = 7$$ Combine like terms: $$-16x - 33 = 7$$ Add 33 to both sides: $$-16x = 40$$ Divide by -16: $$x = \frac{40}{-16} = -\frac{5}{2}$$ Substitute $$x = -\frac{5}{2}$$ into $$y = 6x + 11$$: $$y = 6\left(-\frac{5}{2}\right) + 11$$ $$y = -15 + 11 = -4$$ So the solution is $$\left( -\frac{5}{2}, -4 \right)$$. --- Would you like further details or explanations on any step? --- Here are 5 related practice questions: 1. Solve the system: $$3x + y = 7$$ and $$5x - 2y = 3$$. 2. Verify the solution to the system $$x + 2y = 4$$ and $$3x - y = 5$$. 3. Graph the system $$2x + 3y = 6$$ and $$y = x + 1$$. 4. Solve the system $$4x - 3y = 9$$ and $$y = 2x - 7$$. 5. What is the solution to $$5x + 4y = 12$$ and $$x - y = 3$$? --- **Tip**: When solving systems of equations, substitution works well when one equation is already solved for a variable!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Algebra
Linear Equations
Systems of Equations
Substitution Method
Formulas
y = mx + b
Substitution into linear systems
Theorems
Linear Equation Theorem
Substitution Method in Solving Systems
Suitable Grade Level
Grades 8-10