Math Problem Statement
Which systems of equations have no solutions?
Solution
To determine which systems of equations have no solutions, we need to analyze each option carefully. A system of equations has no solutions when the lines represented by the equations are parallel. Parallel lines have the same slope but different intercepts.
Let’s check each system:
A.
x + 3y = 12 \\ -2x + y = 3 \end{cases}$$ Convert both equations to slope-intercept form ($$y = mx + b$$): 1. From $$x + 3y = 12$$: $$3y = -x + 12 \implies y = -\frac{1}{3}x + 4$$ 2. From $$-2x + y = 3$$: $$y = 2x + 3$$ - Slopes: $$-\frac{1}{3}$$ and $$2$$. The slopes are **different**, so the lines intersect. **This system has a solution.** --- ### B. $$\begin{cases} y = 2x + 7 \\ 6x - 3y = -18 \end{cases}$$ Convert the second equation to slope-intercept form: 1. $$6x - 3y = -18$$: $$-3y = -6x - 18 \implies y = 2x + 6$$ - First equation: $$y = 2x + 7$$. - Second equation: $$y = 2x + 6$$. - Slopes: $$2$$ and $$2$$, which are the **same**. - Intercepts: $$7$$ and $$6$$, which are **different**. The lines are parallel and do not intersect. **This system has no solution.** --- ### C. $$\begin{cases} 9x + 4y = 13 \\ 3x - 6y = 8 \end{cases}$$ Convert both equations to slope-intercept form: 1. From $$9x + 4y = 13$$: $$4y = -9x + 13 \implies y = -\frac{9}{4}x + \frac{13}{4}$$ 2. From $$3x - 6y = 8$$: $$-6y = -3x + 8 \implies y = \frac{1}{2}x - \frac{4}{3}$$ - Slopes: $$-\frac{9}{4}$$ and $$\frac{1}{2}$$. The slopes are **different**, so the lines intersect. **This system has a solution.** --- ### D. $$\begin{cases} x + y = 6 \\ -4x - 4y = -12 \end{cases}$$ Simplify the second equation by dividing through by $$-4$$: $$-4x - 4y = -12 \implies x + y = 3$$ - First equation: $$x + y = 6$$. - Second equation: $$x + y = 3$$. These are **contradictory** because they represent parallel lines with different intercepts. **This system has no solution.** --- ### E. $$\begin{cases} x + y = 20 \\ x - y = 8 \end{cases}$$ Solve this system by substitution or elimination: 1. Add the two equations: $$(x + y) + (x - y) = 20 + 8 \implies 2x = 28 \implies x = 14.$$ 2. Substitute $$x = 14$$ into $$x + y = 20$$: $$14 + y = 20 \implies y = 6.$$ The solution is $$(14, 6)$$. **This system has a solution.** --- ### Final Answer: The systems of equations that have no solutions are: - **B** - **D** --- ### Do you need further clarification on any of these steps? Let me know! Here are 5 related questions to expand your understanding: 1. How can you determine if two lines are parallel using their slopes? 2. What is the geometric interpretation of a system with no solutions? 3. How do you solve a system of equations graphically? 4. How can elimination and substitution methods help solve systems of equations? 5. What is the difference between consistent and inconsistent systems of equations? **Tip**: Always rewrite equations into slope-intercept form ($$y = mx + b$$) to easily compare their slopes and intercepts.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Equations
Slope-Intercept Form
Parallel Lines
Systems of Equations
Formulas
Slope-intercept form: y = mx + b
Equality of slopes for parallel lines: m1 = m2
Theorems
Parallel lines have the same slope but different y-intercepts.
Suitable Grade Level
Grades 8-10