Problem 71
Question
Use the graphing method to tell how many solutions the system has. $$\begin{aligned} 4 x-y &=-2 \\ -12 x+3 y &=6 \end{aligned}$$
Step-by-Step Solution
Verified Answer
From the graph, it can be observed that the two lines coincide with each other, suggesting they are one and the same line. Therefore, the system has infinite solutions.
1Step 1: Express the equations in slope-intercept form
We need to write both equations of the system in slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. The first equation in slope intercept form is \(y = 4x + 2\) and the second equation is \(y = -4x + 2\).
2Step 2: Graph the two lines
Plot the two equations on a graph. The two lines will intersect if there is any solution for the linear system.
3Step 3: Count the number of intersection points
The number of intersection points is the number of solutions of the system. Observe the graph and count how many times the two lines intersect.
Key Concepts
Slope-Intercept FormLinear EquationsIntersection Points of Lines
Slope-Intercept Form
Understanding the slope-intercept form is crucial when dealing with linear equations. It’s a format that allows us to interpret graphically what a linear equation looks like. The general form is given by \(y = mx + b\), where \(m\) represents the slope of the line, and \(b\) indicates the y-intercept, the point where the line crosses the y-axis.
For instance, given a line with the equation \(4x - y = -2\), we can rearrange it to the slope-intercept form by solving for \(y\). Adding \(y\) to both sides and subtracting \(2\) from both gives us \(y = 4x + 2\). The slope here is \(4\), suggesting a relatively steep line, while the y-intercept (\(b\)) is \(2\), marking the spot on the y-axis where our line starts.
Alternatively, for the equation \(-12x + 3y = 6\), dividing the entire equation by \(3\) yields the slope-intercept form \(y = -4x + 2\). Here, the negative slope indicates that the line descends from left to right, and it crosses the y-axis at the same point as the first equation.
For instance, given a line with the equation \(4x - y = -2\), we can rearrange it to the slope-intercept form by solving for \(y\). Adding \(y\) to both sides and subtracting \(2\) from both gives us \(y = 4x + 2\). The slope here is \(4\), suggesting a relatively steep line, while the y-intercept (\(b\)) is \(2\), marking the spot on the y-axis where our line starts.
Alternatively, for the equation \(-12x + 3y = 6\), dividing the entire equation by \(3\) yields the slope-intercept form \(y = -4x + 2\). Here, the negative slope indicates that the line descends from left to right, and it crosses the y-axis at the same point as the first equation.
Linear Equations
A linear equation is an algebraic equation that forms a straight line when graphed on a coordinate plane. These are usually written in the form \(Ax + By = C\), where \(A\), \(B\) and \(C\) are constants. To graph a linear equation, we can rewrite it in the slope-intercept form. This unearths the rate of change or slope, as well as where it touches the y-axis, providing a straightforward way to sketch the graph.
The beauty of linear equations lies in their predictability and the straight line they produce, depicting a constant rate of change. They are foundational in algebra and pave the way for understanding more complex relationships.
The beauty of linear equations lies in their predictability and the straight line they produce, depicting a constant rate of change. They are foundational in algebra and pave the way for understanding more complex relationships.
Intersection Points of Lines
The intersection points of lines represent the solutions to a system of equations. When two lines intersect, the point at which they cross is common to both lines. In the context of linear equations, this single point comprises an ordered pair \((x, y)\) that satisfies both equations simultaneously.
When graphing two linear equations, as seen in our example system above, we visually seek out where the lines intersect. If the lines cross exactly once, there is one unique solution to the system. If they don't cross at all, which happens with parallel lines, there is no solution. If they lie on top of each other, then there are infinitely many solutions, because every point on the lines satisfies both equations.
In our case, the intersection point would be used to determine the solution to the system. The fact that the lines represented by the equations \(y = 4x + 2\) and \(y = -4x + 2\) have the same y-intercept but different slopes suggests they will intersect, providing a single, unique solution.
When graphing two linear equations, as seen in our example system above, we visually seek out where the lines intersect. If the lines cross exactly once, there is one unique solution to the system. If they don't cross at all, which happens with parallel lines, there is no solution. If they lie on top of each other, then there are infinitely many solutions, because every point on the lines satisfies both equations.
In our case, the intersection point would be used to determine the solution to the system. The fact that the lines represented by the equations \(y = 4x + 2\) and \(y = -4x + 2\) have the same y-intercept but different slopes suggests they will intersect, providing a single, unique solution.
Other exercises in this chapter
Problem 71
Divide. $$ 0.5 \div 0.2 $$
View solution Problem 71
Evaluate \(\frac{1.1 \times 10^{-1}}{5.5 \times 10^{-5}}\) $$f. 0.2 \times 10^{-6}$$ $$g. 0.2 \times 10^{-4}$$ $$h. 2 \times 10^{3}$$ $$i. 2 \times 10^{4}$$
View solution Problem 71
Evaluate the expression \(\left(2^{3}\right)^{2}.\) F) 18 G) 32 H) 36 J) 64
View solution Problem 72
Evaluate the expression. Then simplify the answer. $$ \frac{6 \cdot 5}{1+7 \cdot 2} $$
View solution