3. Malik used the method of elimination to solve the system below: $$ \begin{aligned} 4 x+3 y=12 \ 2 x+y=5\end{aligned} $$ $$ \begin{array}{ll}\text { (a) Malik first rewrote the second equation as: } & \text { (b) Malik correctly solves this system using } \ \text { elimination. Was the value he found for } x \ \text { larger or smaller than 1? Justify. }\end{array} $$ What property justifies writing this equation?

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UpStudy AI · Accepted Answer

To solve the problem, we will follow the steps outlined in the prompt.

### Step 1: Rewrite the second equation

The second equation is given as:
$$
2x + y = 5
$$
We can express $$y$$ in terms of $$x$$ by isolating $$y$$:
$$
y = 5 - 2x
$$
This is the rewritten form of the second equation.

### Step 2: Substitute into the first equation

Now, we will substitute this expression for $$y$$ into the first equation:
$$
4x + 3y = 12
$$
Substituting $$y$$:
$$
4x + 3(5 - 2x) = 12
$$

### Step 3: Simplify and solve for $$x$$

Now, we simplify the equation:
$$
4x + 15 - 6x = 12
$$
Combine like terms:
$$
-2x + 15 = 12
$$
Now, isolate $$x$$:
$$
-2x = 12 - 15
$$
$$
-2x = -3
$$
$$
x = \frac{-3}{-2} = \frac{3}{2}
$$

### Step 4: Determine if $$x$$ is larger or smaller than 1

Now we need to determine if the value of $$x$$ is larger or smaller than 1:
$$
\frac{3}{2} = 1.5
$$
Since $$1.5$$ is larger than $$1$$, we conclude that the value Malik found for $$x$$ is larger than $$1$$.

### Step 5: Justification of the property used

The property that justifies rewriting the second equation as $$y = 5 - 2x$$ is the **property of equality**. This property states that if two expressions are equal, you can manipulate one side of the equation (in this case, isolating $$y$$) without changing the equality.

### Summary of the solution

- (a) The second equation was rewritten as $$y = 5 - 2x$$.
- (b) The value of $$x$$ found is $$\frac{3}{2}$$, which is larger than $$1$$.
- The property justifying the rewriting is the property of equality.
Malik first rewrote the second equation as $$ y = 5 - 2x $$. He then substituted this into the first equation and solved for $$ x $$, finding $$ x = \frac{3}{2} $$, which is larger than 1. The property used to rewrite the equation is the property of equality.

Malik used the method of elimination to solve the system below:

What property justifies writing this equation?

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Malik used the method of elimination to solve the system below: What property justifies writing this equation?