Problem 120 Solve graphically and then check... [FREE SOLUTION] (2024)

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Chapter 2: Problem 120

Solve graphically and then check by solving algebraically. $$ 3 x-2=5 x-9 $$

Short Answer

Expert verified

x = \frac{7}{2}, y = \frac{17}{2}

Step by step solution

- Rewrite the equation in slope-intercept form

Begin by rewriting the original equation in a form that is easier to graph. Start by subtracting $3x$ from both sides of the equation to isolate $x$ terms on one side: \[ 3x - 2 - 3x = 5x - 9 - 3x \] This simplifies to: \[ -2 = 2x - 9 \] Next, add 9 to both sides to isolate the $x$ term: \[ -2 + 9 = 2x - 9 + 9 \] which simplifies to: \[ 7 = 2x \] Finally, divide by 2: \[ x = \frac{7}{2} \]

- Graph the equation

Graph the equation using the slope-intercept collected in the previous step: Start by plotting the equation $ y = 3x - 2 $ and $ y = 5x - 9$ on the graph. For $ y = 3x - 2 $: The slope is 3 and y-intercept is -2. For $ y = 5x - 9 $: The slope is 5 and y-intercept is -9. Identify the point where two lines intersect. This intersection point gives the solution for the values of x, y that will satisfy both equations.

- Check graphically

Observe the graph and note the intersection point of the two lines. This intersection represents the solution to the equation. The point of intersection is $( \frac{7}{2}, \frac{17}{2} )$.

- Check algebraically

We already simplified the given equation to derive the result for $x$ as $\frac{7}{2}$. Now, substitute $x$ back into the either of the original equations to find $y$: Utilizing $ y = 3x - 2$ and substituting $x =\frac{7}{2}$: \[ y = 3 \times \frac{7}{2} - 2 \] Simplify the expression: \[ y = \frac{21}{2} - 2 \] Convert $2$, or $\frac{4}{2}$: \[ y = \frac{21}{2} - \frac{4}{2} \] This results in: \[ y = \frac{17}{2} \]

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

slope-intercept form

Understanding the slope-intercept form is crucial when solving linear equations graphically. It is expressed as $ y = mx + b $, where:

$m$ is the slope, representing the rate of change of the line. It tells us how steep the line is.
$b$ is the y-intercept, which is the point where the line crosses the y-axis.

By rewriting an equation in this form, we make it easier to plot on a graph. For example, in the exercise, we derived $ y = 3x - 2 $ and $ y = 5x - 9 $. It's important to note that:

$ y = 3x-2 $ has a slope of 3 and a y-intercept of -2.
$ y = 5x-9 $ has a slope of 5 and a y-intercept of -9.

These values allow us to plot the lines accurately and find their point of intersection.

linear equations

Linear equations are equations of the first degree, meaning they involve only the highest exponent of the variable being 1. They take the general form of:

$ Ax + By = C $, where A, B, and C are constants.

To solve these equations graphically, we often convert them to slope-intercept form first. In the exercise above, we started with $ 3x - 2 = 5x - 9 $. To isolate the variable terms, we performed operations to get to:

$ 7 = 2x $
then simplified to: $ x = \frac{7}{2} $

This shows how algebraic manipulation helps in simplifying and solving linear equations.

graphical solution

A graphical solution involves plotting equations on a graph to find their point of intersection. This solution is visual and helps in understanding the relationship between two linear equations.
In our example, we plotted $ y = 3x - 2 $ and $ y = 5x - 9 $ on a graph. Each line represents one of the equations. Where they intersect tells us the values of $x$ and $y$ that satisfy both equations.
In this case, the point of intersection was found at $\left( \frac{7}{2}, \frac{17}{2} \right) $, confirming our algebraic solution. This method helps verify that the values derived algebraically are accurate.

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Most popular questions from this chapter

Combine like terms. [ 1.3] Solve for $m: x=\frac{m p}{c}$.If $(-10,-2),(-3,4),$ and $(6,4)$ are coordinates of three vertices (corners)of a parallelogram, determine the coordinates of three different points thatcould serve as the fourth vertex.Under what condition(s) will the $x$ - and $y$ -intercepts of a line coincide?What would the equation for such a line look like?Assume that $r, p,$ and $s$ are nonzero constants and that $x$ and $y$ arevariables. Determine whether each equation is linear. $$ p y=s x-r^{2} y-9 $$

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